Hilbert space#quantum mechanics

One of the most familiar examples of a Hilbert space is the Euclidean vector space consisting of three-dimensional vectors, denoted by {{math|R³}}, and equipped with the dot product. The dot product takes two vectors {{math|x}} and {{math|y}}, and produces a real number {{math|x ⋅ y}}. If {{math|x}} and {{math|y}} are represented in Cartesian coordinates, then the dot product is defined by

$$\backslash begin\{pmatrix\}\; x\_1\; \backslash \backslash \; x\_2\; \backslash \backslash \; x\_3\; \backslash end\{pmatrix\}\; \backslash cdot\; \backslash begin\{pmatrix\}\; y\_1\; \backslash \backslash \; y\_2\; \backslash \backslash \; y\_3\; \backslash end\{pmatrix\}\; =\; x\_1\; y\_1\; +\; x\_2\; y\_2\; +\; x\_3\; y\_3\; \backslash ,.$$

The dot product satisfies the properties{{harvnb|Axler|2014|loc=p. 164 §6.2}}

1. It is symmetric in {{math|x}} and {{math|y}}: {{math|1=x ⋅ y = y ⋅ x}}.
2. It is linear in its first argument: {{math|1=(ax₁ + bx₂) ⋅ y = a(x₁ ⋅ y) + b(x₂ ⋅ y)}} for any scalars {{mvar|a}}, {{mvar|b}}, and vectors {{math|x₁}}, {{math|x₂}}, and {{math|y}}.
3. It is positive definite: for all vectors {{math|x}}, {{math|x ⋅ x ≥ 0 }}, with equality if and only if {{math|1= x = 0}}.

An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space. Every finite-dimensional inner product space is also a Hilbert space.However, some sources call finite-dimensional spaces with these properties pre-Hilbert spaces, reserving the term "Hilbert space" for infinite-dimensional spaces; see, e.g., {{harvnb|Levitan|2001}}. The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length (or norm) of a vector, denoted {{math|{{norm|x}}}}, and to the angle {{mvar|θ}} between two vectors {{math|x}} and {{math|y}} by means of the formula

$$\backslash mathbf\{x\}\backslash cdot\backslash mathbf\{y\}\; =\; \backslash left\backslash |\backslash mathbf\{x\}\backslash right\backslash |\; \backslash left\backslash |\backslash mathbf\{y\}\backslash right\backslash |\; \backslash ,\; \backslash cos\backslash theta\; \backslash ,.$$

File:Completeness in Hilbert space.png net displacement vector (in orange).]]

Multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist. A mathematical series

$$\backslash sum\_\{n=0\}^\backslash infty\; \backslash mathbf\{x\}\_n$$

consisting of vectors in {{math|R³}} is absolutely convergent provided that the sum of the lengths converges as an ordinary series of real numbers:{{harvnb|Marsden|1974|loc=§2.8}}

Just as with a series of scalars, a series of vectors that converges absolutely also converges to some limit vector {{math|L}} in the Euclidean space, in the sense that

$$\backslash Biggl\backslash |\; \backslash mathbf\{L\}\; -\; \backslash sum\_\{k=0\}^N\; \backslash mathbf\{x\}\_k\; \backslash Biggr\backslash |\; \backslash to\; 0\; \backslash quad\; \backslash text\{as\; \}\; N\; \backslash to\backslash infty\; \backslash ,.$$

This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense.

Hilbert spaces are often taken over the complex numbers. The complex plane denoted by {{math|C}} is equipped with a notion of magnitude, the complex modulus {{math|{{abs|z}}}}, which is defined as the square root of the product of {{mvar|z}} with its complex conjugate:

$$|z|^2\; =\; z\backslash overline\{z\}\; \backslash ,.$$

If {{math|1=z = x + iy}} is a decomposition of {{mvar|z}} into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length:

$$|z|\; =\; \backslash sqrt\{x^2\; +\; y^2\}\; \backslash ,.$$

The inner product of a pair of complex numbers {{mvar|z}} and {{mvar|w}} is the product of {{mvar|z}} with the complex conjugate of {{mvar|w}}:

$$\backslash langle\; z,\; w\backslash rangle\; =\; z\backslash overline\{w\}\backslash ,.$$

This is complex-valued. The real part of {{math|⟨z, w⟩}} gives the usual two-dimensional Euclidean dot product.

A second example is the space {{math|C²}} whose elements are pairs of complex numbers {{math|1=z = (z₁, z₂)}}. Then an inner product of {{mvar|z}} with another such vector {{math|1=w = (w₁, w₂)}} is given by

$$\backslash langle\; z,\; w\backslash rangle\; =\; z\_1\backslash overline\{w\_1\}\; +\; z\_2\backslash overline\{w\_2\}\backslash ,.$$

The real part of {{math|⟨z, w⟩}} is then the four-dimensional Euclidean dot product. This inner product is Hermitian symmetric, which means that the result of interchanging {{mvar|z}} and {{mvar|w}} is the complex conjugate:

$$\backslash langle\; w,\; z\backslash rangle\; =\; \backslash overline\{\backslash langle\; z,\; w\backslash rangle\}\backslash ,.$$

A {{em|Hilbert space}} is a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product.The mathematical material in this section can be found in any good textbook on functional analysis, such as {{harvtxt|Dieudonné|1960}}, {{harvtxt|Hewitt|Stromberg|1965}}, {{harvtxt|Reed|Simon|1980}} or {{harvtxt|Rudin|1987}}.

To say that a complex vector space {{math|H}} is a {{em|complex inner product space}} means that there is an inner product $\backslash langle\; x,\; y\; \backslash rangle$ associating a complex number to each pair of elements $x,\; y$ of {{math|H}} that satisfies the following properties:

1. The inner product is conjugate symmetric; that is, the inner product of a pair of elements is equal to the complex conjugate of the inner product of the swapped elements: $$\backslash langle\; y,\; x\backslash rangle\; =\; \backslash overline\{\backslash langle\; x,\; y\backslash rangle\}\backslash ,.$$ Importantly, this implies that $\backslash langle\; x,\; x\backslash rangle$ is a real number.
2. The inner product is linear in its firstIn some conventions, inner products are linear in their second arguments instead. argument. For all complex numbers $a$ and $b,$ $$\backslash langle\; ax\_1\; +\; bx\_2,\; y\backslash rangle\; =\; a\backslash langle\; x\_1,\; y\backslash rangle\; +\; b\backslash langle\; x\_2,\; y\backslash rangle\backslash ,.$$
3. The inner product of an element with itself is positive definite: $$\backslash begin\{alignat\}\{4\}$$

\langle x, x\rangle > 0 & \quad \text{ if } x \neq 0, \\

\langle x, x\rangle = 0 & \quad \text{ if } x = 0\,.

\end{alignat}

It follows from properties 1 and 2 that a complex inner product is {{em|antilinear}}, also called {{em|conjugate linear}}, in its second argument, meaning that

$$\backslash langle\; x,\; ay\_1\; +\; by\_2\backslash rangle\; =\; \backslash bar\{a\}\backslash langle\; x,\; y\_1\backslash rangle\; +\; \backslash bar\{b\}\backslash langle\; x,\; y\_2\backslash rangle\backslash ,.$$

A {{em|real inner product space}} is defined in the same way, except that {{math|H}} is a real vector space and the inner product takes real values. Such an inner product will be a bilinear map and $(H,\; H,\; \backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle)$ will form a dual system.{{sfn|ps=|Schaefer|Wolff|1999|pp=122-202}}

The norm is the real-valued function

$$\backslash |x\backslash |\; =\; \backslash sqrt\{\backslash langle\; x,\; x\; \backslash rangle\}\backslash ,,$$

and the distance $d$ between two points $x,\; y$ in {{math|H}} is defined in terms of the norm by

$$d(x,\; y)\; =\; \backslash |x\; -\; y\backslash |\; =\; \backslash sqrt\{\backslash langle\; x\; -\; y,\; x\; -\; y\; \backslash rangle\}\backslash ,.$$

That this function is a distance function means firstly that it is symmetric in $x$ and $y,$ secondly that the distance between $x$ and itself is zero, and otherwise the distance between $x$ and $y$ must be positive, and lastly that the triangle inequality holds, meaning that the length of one leg of a triangle {{math|xyz}} cannot exceed the sum of the lengths of the other two legs:

$$d(x,\; z)\; \backslash leq\; d(x,\; y)\; +\; d(y,\; z)\backslash ,.$$

: File:Triangle inequality in a metric space.svg

This last property is ultimately a consequence of the more fundamental Cauchy–Schwarz inequality, which asserts

$$\backslash left|\backslash langle\; x,\; y\backslash rangle\backslash right|\; \backslash leq\; \backslash |x\backslash |\; \backslash |y\backslash |$$

with equality if and only if $x$ and $y$ are linearly dependent.

With a distance function defined in this way, any inner product space is a metric space, and sometimes is known as a {{em|pre-Hilbert space}}.{{harvnb|Dieudonné|1960|loc=§6.2}} Any pre-Hilbert space that is additionally also a complete space is a Hilbert space.{{harvnb|Roman|2008|loc=p. 327}}

The {{em|completeness}} of {{math|H}} is expressed using a form of the Cauchy criterion for sequences in {{math|H}}: a pre-Hilbert space {{math|H}} is complete if every Cauchy sequence converges with respect to this norm to an element in the space. Completeness can be characterized by the following equivalent condition: if a series of vectors

$$\backslash sum\_\{k=0\}^\backslash infty\; u\_k$$

converges absolutely in the sense that

then the series converges in {{math|H}}, in the sense that the partial sums converge to an element of {{math|H}}.{{harvnb|Roman|2008|loc=p. 330 Theorem 13.8}}

As a complete normed space, Hilbert spaces are by definition also Banach spaces. As such they are topological vector spaces, in which topological notions like the openness and closedness of subsets are well defined. Of special importance is the notion of a closed linear subspace of a Hilbert space that, with the inner product induced by restriction, is also complete (being a closed set in a complete metric space) and therefore a Hilbert space in its own right.

The sequence space {{math|l²}} consists of all infinite sequences {{math|1=z = (z₁, z₂, ...)}} of complex numbers such that the following series converges:{{harvnb|Stein|Shakarchi|2005|p=163}}

$$\backslash sum\_\{n=1\}^\backslash infty\; |z\_n|^2$$

The inner product on {{math|l²}} is defined by:

$$\backslash langle\; \backslash mathbf\{z\},\; \backslash mathbf\{w\}\backslash rangle\; =\; \backslash sum\_\{n=1\}^\backslash infty\; z\_n\backslash overline\{w\_n\}\backslash ,,$$

This second series converges as a consequence of the Cauchy–Schwarz inequality and the convergence of the previous series.

Completeness of the space holds provided that whenever a series of elements from {{math|l²}} converges absolutely (in norm), then it converges to an element of {{math|l²}}. The proof is basic in mathematical analysis, and permits mathematical series of elements of the space to be manipulated with the same ease as series of complex numbers (or vectors in a finite-dimensional Euclidean space).{{harvnb|Dieudonné|1960}}

File:Hilbert.jpg]]

Prior to the development of Hilbert spaces, other generalizations of Euclidean spaces were known to mathematicians and physicists. In particular, the idea of an abstract linear space (vector space) had gained some traction towards the end of the 19th century:Largely from the work of Hermann Grassmann, at the urging of August Ferdinand Möbius {{harv|Boyer|Merzbach|1991|pp=584–586}}. The first modern axiomatic account of abstract vector spaces ultimately appeared in Giuseppe Peano's 1888 account ({{harvnb|Grattan-Guinness|2000|loc=§5.2.2}}; {{harvnb|O'Connor|Robertson|1996}}). this is a space whose elements can be added together and multiplied by scalars (such as real or complex numbers) without necessarily identifying these elements with "geometric" vectors, such as position and momentum vectors in physical systems. Other objects studied by mathematicians at the turn of the 20th century, in particular spaces of sequences (including series) and spaces of functions,A detailed account of the history of Hilbert spaces can be found in {{harvnb|Bourbaki|1987}}. can naturally be thought of as linear spaces. Functions, for instance, can be added together or multiplied by constant scalars, and these operations obey the algebraic laws satisfied by addition and scalar multiplication of spatial vectors.

In the first decade of the 20th century, parallel developments led to the introduction of Hilbert spaces. The first of these was the observation, which arose during David Hilbert and Erhard Schmidt's study of integral equations,{{harvnb|Schmidt|1908}} that two square-integrable real-valued functions {{mvar|f}} and {{mvar|g}} on an interval {{math|[a, b]}} have an inner product

: $\backslash langle\; f,\; g\; \backslash rangle\; =\; \backslash int\_a^b\; f(x)g(x)\backslash ,\; \backslash mathrm\{d\}x$

that has many of the familiar properties of the Euclidean dot product. In particular, the idea of an orthogonal family of functions has meaning. Schmidt exploited the similarity of this inner product with the usual dot product to prove an analog of the spectral decomposition for an operator of the form

: $f(x)\; \backslash mapsto\; \backslash int\_a^b\; K(x,\; y)\; f(y)\backslash ,\; \backslash mathrm\{d\}y$

where {{mvar|K}} is a continuous function symmetric in {{mvar|x}} and {{mvar|y}}. The resulting eigenfunction expansion expresses the function {{mvar|K}} as a series of the form

: $K(x,\; y)\; =\; \backslash sum\_n\; \backslash lambda\_n\backslash varphi\_n(x)\backslash varphi\_n(y)$

where the functions {{mvar|φ_n}} are orthogonal in the sense that {{math|1=⟨φ_n, φ_m⟩ = 0}} for all {{math|n ≠ m}}. The individual terms in this series are sometimes referred to as elementary product solutions. However, there are eigenfunction expansions that fail to converge in a suitable sense to a square-integrable function: the missing ingredient, which ensures convergence, is completeness.{{harvnb|Titchmarsh|1946|loc=§IX.1}}

The second development was the Lebesgue integral, an alternative to the Riemann integral introduced by Henri Lebesgue in 1904.{{harvnb|Lebesgue|1904}}. Further details on the history of integration theory can be found in {{harvtxt|Bourbaki|1987}} and {{harvtxt|Saks|2005}}. The Lebesgue integral made it possible to integrate a much broader class of functions. In 1907, Frigyes Riesz and Ernst Sigismund Fischer independently proved that the space {{math|L²}} of square Lebesgue-integrable functions is a complete metric space.{{harvnb|Bourbaki|1987}} As a consequence of the interplay between geometry and completeness, the 19th century results of Joseph Fourier, Friedrich Bessel and Marc-Antoine Parseval on trigonometric series easily carried over to these more general spaces, resulting in a geometrical and analytical apparatus now usually known as the Riesz–Fischer theorem.{{harvnb|Dunford|Schwartz|1958|loc=§IV.16}}

Further basic results were proved in the early 20th century. For example, the Riesz representation theorem was independently established by Maurice Fréchet and Frigyes Riesz in 1907.In {{harvtxt|Dunford|Schwartz|1958|loc=§IV.16}}, the result that every linear functional on {{math|L²[0,1]}} is represented by integration is jointly attributed to {{harvtxt|Fréchet|1907}} and {{harvtxt|Riesz|1907}}. The general result, that the dual of a Hilbert space is identified with the Hilbert space itself, can be found in {{harvtxt|Riesz|1934}}. John von Neumann coined the term abstract Hilbert space in his work on unbounded Hermitian operators.{{harvnb|von Neumann|1929}} Although other mathematicians such as Hermann Weyl and Norbert Wiener had already studied particular Hilbert spaces in great detail, often from a physically motivated point of view, von Neumann gave the first complete and axiomatic treatment of them.{{harvnb|Kline|1972|p=1092}} Von Neumann later used them in his seminal work on the foundations of quantum mechanics,{{harvnb|Hilbert|Nordheim|von Neumann|1927}} and in his continued work with Eugene Wigner. The name "Hilbert space" was soon adopted by others, for example by Hermann Weyl in his book on quantum mechanics and the theory of groups.{{harvnb|Weyl|1931}}

The significance of the concept of a Hilbert space was underlined with the realization that it offers one of the best mathematical formulations of quantum mechanics.{{harvnb|Prugovečki|1981|pp=1–10}} In short, the states of a quantum mechanical system are vectors in a certain Hilbert space, the observables are hermitian operators on that space, the symmetries of the system are unitary operators, and measurements are orthogonal projections. The relation between quantum mechanical symmetries and unitary operators provided an impetus for the development of the unitary representation theory of groups, initiated in the 1928 work of Hermann Weyl. On the other hand, in the early 1930s it became clear that classical mechanics can be described in terms of Hilbert space (Koopman–von Neumann classical mechanics) and that certain properties of classical dynamical systems can be analyzed using Hilbert space techniques in the framework of ergodic theory.{{harvnb|von Neumann|1932}}

The algebra of observables in quantum mechanics is naturally an algebra of operators defined on a Hilbert space, according to Werner Heisenberg's matrix mechanics formulation of quantum theory.{{harvnb|Peres|1993|pp=79–99}} Von Neumann began investigating operator algebras in the 1930s, as rings of operators on a Hilbert space. The kind of algebras studied by von Neumann and his contemporaries are now known as von Neumann algebras.{{harvnb|Murphy|1990|p=112}} In the 1940s, Israel Gelfand, Mark Naimark and Irving Segal gave a definition of a kind of operator algebras called C*-algebras that on the one hand made no reference to an underlying Hilbert space, and on the other extrapolated many of the useful features of the operator algebras that had previously been studied. The spectral theorem for self-adjoint operators in particular that underlies much of the existing Hilbert space theory was generalized to C*-algebras.{{harvnb|Murphy|1990|p=72}} These techniques are now basic in abstract harmonic analysis and representation theory.

{{Main|Lp space|l1={{mvar|L^p}} space}}

Lebesgue spaces are function spaces associated to measure spaces {{math|(X, M, μ)}}, where {{math|X}} is a set, {{math|M}} is a σ-algebra of subsets of {{math|X}}, and {{math|μ}} is a countably additive measure on {{math|M}}. Let {{math|L²(X, μ)}} be the space of those complex-valued measurable functions on {{math|X}} for which the Lebesgue integral of the square of the absolute value of the function is finite, i.e., for a function {{math|f}} in {{math|L²(X, μ)}},

and where functions are identified if and only if they differ only on a set of measure zero.

The inner product of functions {{math|f}} and {{math|g}} in {{math|L²(X, μ)}} is then defined as

$$\backslash langle\; f,\; g\backslash rangle\; =\; \backslash int\_X\; f(t)\; \backslash overline\{g(t)\}\; \backslash ,\; \backslash mathrm\{d\}\; \backslash mu(t)$$ or $$\backslash langle\; f,\; g\backslash rangle\; =\; \backslash int\_X\; \backslash overline\{f(t)\}\; g(t)\; \backslash ,\; \backslash mathrm\{d\}\; \backslash mu(t)\; \backslash ,,$$

where the second form (conjugation of the first element) is commonly found in the theoretical physics literature. For {{math|f}} and {{math|g}} in {{math|L²}}, the integral exists because of the Cauchy–Schwarz inequality, and defines an inner product on the space. Equipped with this inner product, {{math|L²}} is in fact complete.{{harvnb|Halmos|1957|loc=Section 42}} The Lebesgue integral is essential to ensure completeness: on domains of real numbers, for instance, not enough functions are Riemann integrable.{{harvnb|Hewitt|Stromberg|1965}}

The Lebesgue spaces appear in many natural settings. The spaces {{math|L²(R)}} and {{math|L²([0,1])}} of square-integrable functions with respect to the Lebesgue measure on the real line and unit interval, respectively, are natural domains on which to define the Fourier transform and Fourier series. In other situations, the measure may be something other than the ordinary Lebesgue measure on the real line. For instance, if {{math|w}} is any positive measurable function, the space of all measurable functions {{math|f}} on the interval {{math|[0, 1]}} satisfying

is called the Lp space#Weighted Lp spaces {{math|L{{su|p=2|b=w|lh=0.8em}}([0, 1])}}, and {{math|w}} is called the weight function. The inner product is defined by

$$\backslash langle\; f,\; g\backslash rangle\; =\; \backslash int\_0^1\; f(t)\; \backslash overline\{g(t)\}\; w(t)\; \backslash ,\; \backslash mathrm\{d\}t\; \backslash ,.$$

The weighted space {{math|L{{su|p=2|b=w|lh=0.8em}}([0, 1])}} is identical with the Hilbert space {{math|L²([0, 1], μ)}} where the measure {{math|μ}} of a Lebesgue-measurable set {{math|A}} is defined by

$$\backslash mu(A)\; =\; \backslash int\_A\; w(t)\backslash ,\backslash mathrm\{d\}t\; \backslash ,.$$

Weighted {{math|L²}} spaces like this are frequently used to study orthogonal polynomials, because different families of orthogonal polynomials are orthogonal with respect to different weighting functions.{{Abramowitz_Stegun_ref|22|773}}

Sobolev spaces, denoted by {{math|H{{i sup|s}}}} or {{math|W{{i sup|s,2}}}}, are Hilbert spaces. These are a special kind of function space in which differentiation may be performed, but that (unlike other Banach spaces such as the Hölder spaces) support the structure of an inner product. Because differentiation is permitted, Sobolev spaces are a convenient setting for the theory of partial differential equations. They also form the basis of the theory of direct methods in the calculus of variations.{{harvnb|Giusti|2003}}

For {{math|s}} a non-negative integer and {{math|Ω ⊂ Rⁿ}}, the Sobolev space {{math|H^s(Ω)}} contains {{math|L²}} functions whose weak derivatives of order up to {{math|s}} are also {{math|L²}}. The inner product in {{math|H^s(Ω)}} is

$$\backslash langle\; f,\; g\backslash rangle\; =\; \backslash int\_\backslash Omega\; f(x)\backslash bar\{g\}(x)\backslash ,\backslash mathrm\{d\}x\; +\; \backslash int\_\backslash Omega\; D\; f(x)\backslash cdot\; D\backslash bar\{g\}(x)\backslash ,\backslash mathrm\{d\}x\; +\; \backslash cdots\; +\; \backslash int\_\backslash Omega\; D^s\; f(x)\backslash cdot\; D^s\; \backslash bar\{g\}(x)\backslash ,\; \backslash mathrm\{d\}x$$

where the dot indicates the dot product in the Euclidean space of partial derivatives of each order. Sobolev spaces can also be defined when {{math|s}} is not an integer.

Sobolev spaces are also studied from the point of view of spectral theory, relying more specifically on the Hilbert space structure. If {{math|Ω}} is a suitable domain, then one can define the Sobolev space {{math|H^s(Ω)}} as the space of Bessel potentials;{{harvnb|Stein|1970}} roughly,

$$H^s(\backslash Omega)\; =\; \backslash left\backslash \{\; (1-\backslash Delta)^\{-s/2\}f\; \backslash mathrel\{\backslash Big|\}\; f\backslash in\; L^2(\backslash Omega)\backslash right\backslash \}\; \backslash ,.$$

Here {{math|Δ}} is the Laplacian and {{math|(1 − Δ)^−s / 2}} is understood in terms of the spectral mapping theorem. Apart from providing a workable definition of Sobolev spaces for non-integer {{math|s}}, this definition also has particularly desirable properties under the Fourier transform that make it ideal for the study of pseudodifferential operators. Using these methods on a compact Riemannian manifold, one can obtain for instance the Hodge decomposition, which is the basis of Hodge theory.Details can be found in {{harvtxt|Warner|1983}}.

The Hardy spaces are function spaces, arising in complex analysis and harmonic analysis, whose elements are certain holomorphic functions in a complex domain.A general reference on Hardy spaces is the book {{harvtxt|Duren|1970}}. Let {{math|U}} denote the unit disc in the complex plane. Then the Hardy space {{math|H²(U)}} is defined as the space of holomorphic functions {{math|f}} on {{math|U}} such that the means

$$M\_r(f)\; =\; \backslash frac\{1\}\{2\backslash pi\}\; \backslash int\_0^\{2\backslash pi\}\; \backslash left|f\backslash bigl(re^\{i\backslash theta\}\backslash bigr)\backslash right|^2\; \backslash ,\; \backslash mathrm\{d\}\backslash theta$$

remain bounded for {{math|r < 1}}. The norm on this Hardy space is defined by

$$\backslash left\backslash |f\backslash right\backslash |\_2\; =\; \backslash lim\_\{r\; \backslash to\; 1\}\; \backslash sqrt\{M\_r(f)\}\; \backslash ,.$$

Hardy spaces in the disc are related to Fourier series. A function {{math|f}} is in {{math|H²(U)}} if and only if

$$f(z)\; =\; \backslash sum\_\{n=0\}^\backslash infty\; a\_n\; z^n$$

where

Thus {{math|H²(U)}} consists of those functions that are L² on the circle, and whose negative frequency Fourier coefficients vanish.

The Bergman spaces are another family of Hilbert spaces of holomorphic functions.{{harvnb|Krantz|2002|loc=§1.4}} Let {{math|D}} be a bounded open set in the complex plane (or a higher-dimensional complex space) and let {{math|L^{2, h}(D)}} be the space of holomorphic functions {{math|f}} in {{math|D}} that are also in {{math|L²(D)}} in the sense that

where the integral is taken with respect to the Lebesgue measure in {{math|D}}. Clearly {{math|L^{2, h}(D)}} is a subspace of {{math|L²(D)}}; in fact, it is a closed subspace, and so a Hilbert space in its own right. This is a consequence of the estimate, valid on compact subsets {{math|K}} of {{math|D}}, that

$$\backslash sup\_\{z\backslash in\; K\}\; \backslash left|f(z)\backslash right|\; \backslash le\; C\_K\; \backslash left\backslash |f\backslash right\backslash |\_2\; \backslash ,,$$

which in turn follows from Cauchy's integral formula. Thus convergence of a sequence of holomorphic functions in {{math|L²(D)}} implies also compact convergence, and so the limit function is also holomorphic. Another consequence of this inequality is that the linear functional that evaluates a function {{math|f}} at a point of {{math|D}} is actually continuous on {{math|L^2,h(D)}}. The Riesz representation theorem implies that the evaluation functional can be represented as an element of {{math|L^2,h(D)}}. Thus, for every {{math|z ∈ D}}, there is a function {{math|η_z ∈ L^2,h(D)}} such that

$$f(z)\; =\; \backslash int\_D\; f(\backslash zeta)\backslash overline\{\backslash eta\_z(\backslash zeta)\}\backslash ,\backslash mathrm\{d\}\backslash mu(\backslash zeta)$$

for all {{math|f ∈ L^2,h(D)}}. The integrand

$$K(\backslash zeta,\; z)\; =\; \backslash overline\{\backslash eta\_z(\backslash zeta)\}$$

is known as the Bergman kernel of {{math|D}}. This integral kernel satisfies a reproducing property

$$f(z)\; =\; \backslash int\_D\; f(\backslash zeta)K(\backslash zeta,\; z)\backslash ,\backslash mathrm\{d\}\backslash mu(\backslash zeta)\; \backslash ,.$$

A Bergman space is an example of a reproducing kernel Hilbert space, which is a Hilbert space of functions along with a kernel {{math|K(ζ, z)}} that verifies a reproducing property analogous to this one. The Hardy space {{math|H²(D)}} also admits a reproducing kernel, known as the Szegő kernel.{{harvnb|Krantz|2002|loc=§1.5}} Reproducing kernels are common in other areas of mathematics as well. For instance, in harmonic analysis the Poisson kernel is a reproducing kernel for the Hilbert space of square-integrable harmonic functions in the unit ball. That the latter is a Hilbert space at all is a consequence of the mean value theorem for harmonic functions.

Many of the applications of Hilbert spaces exploit the fact that Hilbert spaces support generalizations of simple geometric concepts like projection and change of basis from their usual finite dimensional setting. In particular, the spectral theory of continuous self-adjoint linear operators on a Hilbert space generalizes the usual spectral decomposition of a matrix, and this often plays a major role in applications of the theory to other areas of mathematics and physics.

{{Main|Sturm–Liouville theory|Spectral theory of ordinary differential equations}}

File:Harmonic partials on strings.svgs of a vibrating string. These are eigenfunctions of an associated Sturm–Liouville problem. The eigenvalues 1, {{sfrac|1|2}}, {{sfrac|1|3}}, ... form the (musical) harmonic series.]]

In the theory of ordinary differential equations, spectral methods on a suitable Hilbert space are used to study the behavior of eigenvalues and eigenfunctions of differential equations. For example, the Sturm–Liouville problem arises in the study of the harmonics of waves in a violin string or a drum, and is a central problem in ordinary differential equations.{{harvnb|Young|1988|loc=Chapter 9}} The problem is a differential equation of the form

$$-\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}x\}\backslash left[p(x)\backslash frac\{\backslash mathrm\{d\}y\}\{\backslash mathrm\{d\}x\}\backslash right]\; +\; q(x)y\; =\; \backslash lambda\; w(x)y$$

for an unknown function {{math|y}} on an interval {{closed-closed|a, b}}, satisfying general homogeneous Robin boundary conditions

$$\backslash begin\{cases\}$$

\alpha y(a)+\alpha' y'(a) &= 0 \\

\beta y(b) + \beta' y'(b) &= 0 \,.

\end{cases}

The functions {{math|p}}, {{math|q}}, and {{math|w}} are given in advance, and the problem is to find the function {{math|y}} and constants {{math|λ}} for which the equation has a solution. The problem only has solutions for certain values of {{math|λ}}, called eigenvalues of the system, and this is a consequence of the spectral theorem for compact operators applied to the integral operator defined by the Green's function for the system. Furthermore, another consequence of this general result is that the eigenvalues {{math|λ}} of the system can be arranged in an increasing sequence tending to infinity.{{harvnb|Pedersen|1995|loc=§4.4}}The eigenvalues of the Fredholm kernel are {{math|{{sfrac|1|λ}}}}, which tend to zero.

Hilbert spaces form a basic tool in the study of partial differential equations.{{harvnb|Bers|John|Schechter|1981}} For many classes of partial differential equations, such as linear elliptic equations, it is possible to consider a generalized solution (known as a weak solution) by enlarging the class of functions. Many weak formulations involve the class of Sobolev functions, which is a Hilbert space. A suitable weak formulation reduces to a geometrical problem, the analytic problem of finding a solution or, often what is more important, showing that a solution exists and is unique for given boundary data. For linear elliptic equations, one geometrical result that ensures unique solvability for a large class of problems is the Lax–Milgram theorem. This strategy forms the rudiment of the Galerkin method (a finite element method) for numerical solution of partial differential equations.More detail on finite element methods from this point of view can be found in {{harvtxt|Brenner|Scott|2005}}.

A typical example is the Poisson equation {{math|1=−Δu = g}} with Dirichlet boundary conditions in a bounded domain {{math|Ω}} in {{math|R²}}. The weak formulation consists of finding a function {{math|u}} such that, for all continuously differentiable functions {{math|v}} in {{math|Ω}} vanishing on the boundary:

$$\backslash int\_\backslash Omega\; \backslash nabla\; u\backslash cdot\backslash nabla\; v\; =\; \backslash int\_\backslash Omega\; gv\backslash ,.$$

This can be recast in terms of the Hilbert space {{math|H{{su|p=1|b=0}}(Ω)}} consisting of functions {{math|u}} such that {{math|u}}, along with its weak partial derivatives, are square integrable on {{math|Ω}}, and vanish on the boundary. The question then reduces to finding {{math|u}} in this space such that for all {{math|v}} in this space

$$a(u,\; v)\; =\; b(v)$$

where {{math|a}} is a continuous bilinear form, and {{math|b}} is a continuous linear functional, given respectively by

$$a(u,\; v)\; =\; \backslash int\_\backslash Omega\; \backslash nabla\; u\backslash cdot\backslash nabla\; v,\backslash quad\; b(v)=\; \backslash int\_\backslash Omega\; gv\backslash ,.$$

Since the Poisson equation is elliptic, it follows from Poincaré's inequality that the bilinear form {{math|a}} is coercive. The Lax–Milgram theorem then ensures the existence and uniqueness of solutions of this equation.{{harvnb|Brezis|2010}}, section 9.5

Hilbert spaces allow for many elliptic partial differential equations to be formulated in a similar way, and the Lax–Milgram theorem is then a basic tool in their analysis. With suitable modifications, similar techniques can be applied to parabolic partial differential equations and certain hyperbolic partial differential equations.{{harvnb|Evans|1998}}

File:BunimovichStadium.svg ball in the Bunimovich stadium is described by an ergodic dynamical system.]]

The field of ergodic theory is the study of the long-term behavior of chaotic dynamical systems. The protypical case of a field that ergodic theory applies to is thermodynamics, in which—though the microscopic state of a system is extremely complicated (it is impossible to understand the ensemble of individual collisions between particles of matter)—the average behavior over sufficiently long time intervals is tractable. The laws of thermodynamics are assertions about such average behavior. In particular, one formulation of the zeroth law of thermodynamics asserts that over sufficiently long timescales, the only functionally independent measurement that one can make of a thermodynamic system in equilibrium is its total energy, in the form of temperature.{{harvtxt|Pathria|1996}}, Chapters 2 and 3

An ergodic dynamical system is one for which, apart from the energy—measured by the Hamiltonian—there are no other functionally independent conserved quantities on the phase space. More explicitly, suppose that the energy {{math|E}} is fixed, and let {{math|Ω_E}} be the subset of the phase space consisting of all states of energy {{math|E}} (an energy surface), and let {{math|T_t}} denote the evolution operator on the phase space. The dynamical system is ergodic if every invariant measurable functions on {{math|Ω_E}} is constant almost everywhere.{{harvtxt|Einsiedler|Ward|2011}}, Proposition 2.14. An invariant function {{math|f}} is one for which

$$f(T\_tw)\; =\; f(w)$$

for all {{math|w}} on {{math|Ω_E}} and all time {{math|t}}. Liouville's theorem implies that there exists a measure {{math|μ}} on the energy surface that is invariant under the time translation. As a result, time translation is a unitary transformation of the Hilbert space {{math|L²(Ω_E, μ)}} consisting of square-integrable functions on the energy surface {{math|Ω_E}} with respect to the inner product

$$\backslash left\backslash langle\; f,\; g\backslash right\backslash rangle\_\{L^2\backslash left(\backslash Omega\_E,\; \backslash mu\backslash right)\}\; =\; \backslash int\_E\; f\backslash bar\{g\}\backslash ,\backslash mathrm\{d\}\backslash mu\backslash ,.$$

The von Neumann mean ergodic theorem states the following:

• If {{math|U_t}} is a (strongly continuous) one-parameter semigroup of unitary operators on a Hilbert space {{math|H}}, and {{math|P}} is the orthogonal projection onto the space of common fixed points of {{math|U_t}}, {{math|1={x ∈H {{!}} U_tx = x, ∀t > 0}}}, then $$Px\; =\; \backslash lim\_\{T\backslash to\backslash infty\}\; \backslash frac\{1\}\{T\}\; \backslash int\_0^T\; U\_tx\backslash ,\backslash mathrm\{d\}t\backslash ,.$$

For an ergodic system, the fixed set of the time evolution consists only of the constant functions, so the ergodic theorem implies the following:{{harvnb|Reed|Simon|1980}} for any function {{math|f ∈ L²(Ω_E, μ)}},

$$\backslash underset\{T\backslash to\backslash infty\}\{L^2\; -\; \backslash lim\}\; \backslash frac\{1\}\{T\}\backslash int\_0^T\; f(T\_tw)\backslash ,\backslash mathrm\{d\}t\; =\; \backslash int\_\{\backslash Omega\_E\}\; f(y)\backslash ,\backslash mathrm\{d\}\backslash mu(y)\backslash ,.$$

That is, the long time average of an observable {{math|f}} is equal to its expectation value over an energy surface.

File:Sawtooth Fourier Analysys.svg

File:Harmoniki.png, an orthonormal basis for the Hilbert space of square-integrable functions on the sphere, shown graphed along the radial direction]]

One of the basic goals of Fourier analysis is to decompose a function into a (possibly infinite) linear combination of given basis functions: the associated Fourier series. The classical Fourier series associated to a function {{math|f}} defined on the interval {{math|[0, 1]}} is a series of the form

$$\backslash sum\_\{n=-\backslash infty\}^\backslash infty\; a\_n\; e^\{2\backslash pi\; in\backslash theta\}$$

where

$$a\_n\; =\; \backslash int\_0^1f(\backslash theta)\backslash ;\backslash !e^\{-2\backslash pi\; in\backslash theta\}\backslash ,\backslash mathrm\{d\}\backslash theta\backslash ,.$$

The example of adding up the first few terms in a Fourier series for a sawtooth function is shown in the figure. The basis functions are sine waves with wavelengths {{math|{{sfrac|λ|n}}}} (for integer {{math|n}}) shorter than the wavelength {{math|λ}} of the sawtooth itself (except for {{math|1=n = 1}}, the fundamental wave).

A significant problem in classical Fourier series asks in what sense the Fourier series converges, if at all, to the function {{math|f}}. Hilbert space methods provide one possible answer to this question.A treatment of Fourier series from this point of view is available, for instance, in {{harvtxt|Rudin|1987}} or {{harvtxt|Folland|2009}}. The functions {{math|1=e_n(θ) = e^2πinθ}} form an orthogonal basis of the Hilbert space {{math|L²([0, 1])}}. Consequently, any square-integrable function can be expressed as a series

$$f(\backslash theta)\; =\; \backslash sum\_n\; a\_n\; e\_n(\backslash theta)\backslash ,,\backslash quad\; a\_n\; =\; \backslash langle\; f,\; e\_n\backslash rangle$$

and, moreover, this series converges in the Hilbert space sense (that is, in the mean convergence).

The problem can also be studied from the abstract point of view: every Hilbert space has an orthonormal basis, and every element of the Hilbert space can be written in a unique way as a sum of multiples of these basis elements. The coefficients appearing on these basis elements are sometimes known abstractly as the Fourier coefficients of the element of the space.{{harvnb|Halmos|1957|loc=§5}} The abstraction is especially useful when it is more natural to use different basis functions for a space such as {{math|L²([0, 1])}}. In many circumstances, it is desirable not to decompose a function into trigonometric functions, but rather into orthogonal polynomials or wavelets for instance,{{harvnb|Bachman|Narici|Beckenstein|2000}} and in higher dimensions into spherical harmonics.{{harvnb|Stein|Weiss|1971|loc=§IV.2}}

For instance, if {{math|e_n}} are any orthonormal basis functions of {{math|L²[0, 1]}}, then a given function in {{math|L²[0, 1]}} can be approximated as a finite linear combination{{harvnb|Lanczos|1988|pp=212–213}}

$$f(x)\; \backslash approx\; f\_n\; (x)\; =\; a\_1\; e\_1\; (x)\; +\; a\_2\; e\_2(x)\; +\; \backslash cdots\; +\; a\_n\; e\_n\; (x)\backslash ,.$$

The coefficients {{math|{a_j}}} are selected to make the magnitude of the difference {{math|{{norm|f − f_n}}²}} as small as possible. Geometrically, the best approximation is the orthogonal projection of {{math|f}} onto the subspace consisting of all linear combinations of the {{math|{e_j}}}, and can be calculated by{{harvnb|Lanczos|1988|loc=Equation 4-3.10}}

$$a\_j\; =\; \backslash int\_0^1\; \backslash overline\{e\_j(x)\}f\; (x)\; \backslash ,\; \backslash mathrm\{d\}x\backslash ,.$$

That this formula minimizes the difference {{math|{{norm|f − f_n}}²}} is a consequence of Bessel's inequality and Parseval's formula.

In various applications to physical problems, a function can be decomposed into physically meaningful eigenfunctions of a differential operator (typically the Laplace operator): this forms the foundation for the spectral study of functions, in reference to the spectrum of the differential operator.The classic reference for spectral methods is {{harvnb|Courant|Hilbert|1953}}. A more up-to-date account is {{harvnb|Reed|Simon|1975}}. A concrete physical application involves the problem of hearing the shape of a drum: given the fundamental modes of vibration that a drumhead is capable of producing, can one infer the shape of the drum itself?{{harvnb|Kac|1966}} The mathematical formulation of this question involves the Dirichlet eigenvalues of the Laplace equation in the plane, that represent the fundamental modes of vibration in direct analogy with the integers that represent the fundamental modes of vibration of the violin string.

Spectral theory also underlies certain aspects of the Fourier transform of a function. Whereas Fourier analysis decomposes a function defined on a compact set into the discrete spectrum of the Laplacian (which corresponds to the vibrations of a violin string or drum), the Fourier transform of a function is the decomposition of a function defined on all of Euclidean space into its components in the continuous spectrum of the Laplacian. The Fourier transformation is also geometrical, in a sense made precise by the Plancherel theorem, that asserts that it is an isometry of one Hilbert space (the "time domain") with another (the "frequency domain"). This isometry property of the Fourier transformation is a recurring theme in abstract harmonic analysis (since it reflects the conservation of energy for the continuous Fourier Transform), as evidenced for instance by the Plancherel theorem for spherical functions occurring in noncommutative harmonic analysis.

File:HAtomOrbitals.png of an electron in a hydrogen atom are eigenfunctions of the energy.]]

{{main|Measurement in quantum mechanics}}

In the mathematically rigorous formulation of quantum mechanics, developed by John von Neumann,{{harvnb|von Neumann|1955}} the possible states (more precisely, the pure states) of a quantum mechanical system are represented by unit vectors (called state vectors) residing in a complex separable Hilbert space, known as the state space, well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the projectivization of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system; for example, the position and momentum states for a single non-relativistic spin zero particle is the space of all square-integrable functions, while the states for the spin of a single proton are unit elements of the two-dimensional complex Hilbert space of spinors. Each observable is represented by a self-adjoint linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate.{{harvnb|Holevo|2001|p=17}}

The inner product between two state vectors is a complex number known as a probability amplitude. During an ideal measurement of a quantum mechanical system, the probability that a system collapses from a given initial state to a particular eigenstate is given by the square of the absolute value of the probability amplitudes between the initial and final states.{{harvnb|Rieffel|Polak|2011|p=55}} The possible results of a measurement are the eigenvalues of the operator—which explains the choice of self-adjoint operators, for all the eigenvalues must be real. The probability distribution of an observable in a given state can be found by computing the spectral decomposition of the corresponding operator.{{harvnb|Peres|1993|p=101}}

For a general system, states are typically not pure, but instead are represented as statistical mixtures of pure states, or mixed states, given by density matrices: self-adjoint operators of trace one on a Hilbert space.{{harvnb|Peres|1993|pp=73}} Moreover, for general quantum mechanical systems, the effects of a single measurement can influence other parts of a system in a manner that is described instead by a positive operator valued measure. Thus the structure both of the states and observables in the general theory is considerably more complicated than the idealization for pure states.{{harvnb|Nielsen|Chuang|2000|p=90}}

In probability theory, Hilbert spaces also have diverse applications. Here a fundamental Hilbert space is the space of random variables on a given probability space, having class $L^2$ (finite first and second moments). A common operation in statistics is that of centering a random variable by subtracting its expectation. Thus if $X$ is a random variable, then $X\; -\; E(X)$ is its centering. In the Hilbert space view, this is the orthogonal projection of $X$ onto the kernel of the expectation operator, which a continuous linear functional on the Hilbert space (in fact, the inner product with the constant random variable 1), and so this kernel is a closed subspace.

The conditional expectation has a natural interpretation in the Hilbert space.{{harvtxt|Billingsley|1986}}, p. 477, ex. 34.13}} Suppose that a probability space $(\backslash Omega,\; P,\; \backslash mathcal\; B)$ is given, where $\backslash mathcal\; B$ is a sigma algebra on the set $\backslash Omega$, and $P$ is a probability measure on the measure space $(\backslash Omega,\; \backslash mathcal\; B)$. If $\backslash mathcal\; F\backslash le\backslash mathcal\; B$ is a sigma subalgebra of $\backslash mathcal\; B$, then the conditional expectation $E[X|\backslash mathcal\; F]$ is the orthogonal projection of $X$ onto the subspace of $L^2(\backslash Omega,\; P)$ consisting of the $\backslash mathcal\; F$-measurable functions. If the random variable $X$ in $L^2(\backslash Omega,\; P)$ is independent of the sigma algebra $\backslash mathcal\; F$ then conditional expectation $E(X|\backslash mathcal\; F)\; =\; E(X)$, i.e., its projection onto the $\backslash mathcal\; F$-measurable functions is constant. Equivalently, the projection of its centering is zero.

In particular, if two random variables $X$ and $Y$ (in $L^2(\backslash Omega,\; P)$) are independent, then the centered random variables $X-E(X)$ and $Y-E(Y)$ are orthogonal. (This means that the two variables have zero covariance: they are uncorrelated.) In that case, the Pythagorean theorem in the kernel of the expectation operator implies that the variances of $X$ and $Y$ satisfy the identity:

$$\backslash operatorname\{Var\}(X+Y)\; =\; \backslash operatorname\{Var\}(X)\; +\; \backslash operatorname\{Var\}(Y),$$

sometimes called the Pythagorean theorem of statistics, and is of importance in linear regression.{{harvnb|Stapleton|1995}} As {{harvtxt|Stapleton|1995}} puts it, "the analysis of variance may be viewed as the decomposition of the squared length of a vector into the sum of the squared lengths of several vectors, using the Pythagorean Theorem."

The theory of martingales can be formulated in Hilbert spaces. A martingale in a Hilbert space is a sequence $x\_1,x\_2,\backslash dots$ of elements of a Hilbert space such that, for each {{math|n}}, $x\_n$ is the orthogonal projection of $x\_\{n+1\}$ onto the linear hull of $x\_1,\backslash dots,x\_n$.{{harvtxt|Hewitt|Stromberg|1965}}, Exercise 16.45. If the $x\_k$ are random variables, this reproduces the usual definition of a (discrete) martingale: the expectation of $x\_\{n+1\}$, conditioned on $x\_1,\backslash dots,x\_n$, is equal to $x\_n$.

Hilbert spaces are also used throughout the foundations of the Itô calculus.{{harvnb|Karatzas|Shreve|2019}}, Chapter 3 To any square-integrable martingale, it is possible to associate a Hilbert norm on the space of equivalence classes of progressively measurable processes with respect to the martingale (using the quadratic variation of the martingale as the measure). The Itô integral can be constructed by first defining it for simple processes, and then exploiting their density in the Hilbert space. A noteworthy result is then the Itô isometry, which attests that for any martingale M having quadratic variation measure $d\backslash langle\; M\backslash rangle\_t$, and any progressively measurable process H:

$$E\backslash left[\backslash left(\backslash int\_0^tH\_sdM\_s\backslash right)^2\backslash right]\; =\; E\backslash left[\backslash int\_0^tH\_s^2d\backslash langle\; M\backslash rangle\_s\backslash right]$$

whenever the expectation on the right-hand side is finite.

A deeper application of Hilbert spaces that is especially important in the theory of Gaussian processes is an attempt, due to Leonard Gross and others, to make sense of certain formal integrals over infinite dimensional spaces like the Feynman path integral from quantum field theory. The problem with integrals like this is that there is no infinite dimensional Lebesgue measure. The notion of an abstract Wiener space allows one to construct a measure on a Banach space {{math|B}} that contains a Hilbert space {{math|H}}, called the Cameron–Martin space, as a dense subset, out of a finitely additive cylinder set measure on {{math|H}}. The resulting measure on {{math|B}} is countably additive and invariant under translation by elements of {{math|H}}, and this provides a mathematically rigorous way of thinking of the Wiener measure as a Gaussian measure on the Sobolev space $H^1([0,\backslash infty))$.{{harvtxt|Stroock|2011|loc=Chapter 8}}

{{Main|Color vision#Mathematics of color perception}}

Any true physical color can be represented by a combination of pure spectral colors. As physical colors can be composed of any number of spectral colors, the space of physical colors may aptly be represented by a Hilbert space over spectral colors. Humans have three types of cone cells for color perception, so the perceivable colors can be represented by 3-dimensional Euclidean space. The many-to-one linear mapping from the Hilbert space of physical colors to the Euclidean space of human perceivable colors explains why many distinct physical colors may be perceived by humans to be identical (e.g., pure yellow light versus a mix of red and green light, see Metamerism).{{citation|chapter=Mind and nature|author=Hermann Weyl|title=Mind and nature: selected writings on philosophy, mathematics, and physics|year=2009|publisher=Princeton University Press|bibcode=2009mnsw.book.....W }}{{citation|title=Geometry of color perception. Part 2: perceived colors from real quantum states and Hering's rebit|first=M.|last=Berthier|year=2020|journal=The Journal of Mathematical Neuroscience|volume=10|issue=1 |page=14 |doi=10.1186/s13408-020-00092-x |pmid=32902776 |pmc=7481323 |doi-access=free }}

Two vectors {{math|u}} and {{math|v}} in a Hilbert space {{math|H}} are orthogonal when {{math|1=⟨u, v⟩ = 0}}. The notation for this is {{math|u ⊥ v}}. More generally, when {{math|S}} is a subset in {{math|H}}, the notation {{math|u ⊥ S}} means that {{math|u}} is orthogonal to every element from {{math|S}}.

When {{math|u}} and {{math|v}} are orthogonal, one has

$$\backslash |u\; +\; v\backslash |^2\; =\; \backslash langle\; u\; +\; v,\; u\; +\; v\; \backslash rangle\; =\; \backslash langle\; u,\; u\; \backslash rangle\; +\; 2\; \backslash ,\; \backslash operatorname\{Re\}\; \backslash langle\; u,\; v\; \backslash rangle\; +\; \backslash langle\; v,\; v\; \backslash rangle=\; \backslash |u\backslash |^2\; +\; \backslash |v\backslash |^2\backslash ,.$$

By induction on {{math|n}}, this is extended to any family {{math|u₁, ..., u_n}} of {{math|n}} orthogonal vectors,

$$\backslash left\backslash |u\_1\; +\; \backslash cdots\; +\; u\_n\backslash right\backslash |^2\; =\; \backslash left\backslash |u\_1\backslash right\backslash |^2\; +\; \backslash cdots\; +\; \backslash left\backslash |u\_n\backslash right\backslash |^2\; .$$

Whereas the Pythagorean identity as stated is valid in any inner product space, completeness is required for the extension of the Pythagorean identity to series.{{harvnb|Reed|Simon|1980}}, Theorem 12.6 A series {{math|Σu_k}} of orthogonal vectors converges in {{math|H}} if and only if the series of squares of norms converges, and

$$\backslash Biggl\backslash |\backslash sum\_\{k=0\}^\backslash infty\; u\_k\; \backslash Biggr\backslash |^2\; =\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash left\backslash |u\_k\backslash right\backslash |^2\backslash ,.$$

Furthermore, the sum of a series of orthogonal vectors is independent of the order in which it is taken.

File:Color parallelogram.svg

By definition, every Hilbert space is also a Banach space. Furthermore, in every Hilbert space the following parallelogram identity holds:{{harvnb|Reed|Simon|1980}}, p. 38

$$\backslash |u\; +\; v\backslash |^2\; +\; \backslash |u\; -\; v\backslash |^2\; =\; 2\backslash bigl(\backslash |u\backslash |^2\; +\; \backslash |v\backslash |^2\backslash bigr)\backslash ,.$$

Conversely, every Banach space in which the parallelogram identity holds is a Hilbert space, and the inner product is uniquely determined by the norm by the polarization identity.{{harvnb|Young|1988|p=23}} For real Hilbert spaces, the polarization identity is

$$\backslash langle\; u,\; v\backslash rangle\; =\; \backslash tfrac\{1\}\{4\}\backslash bigl(\backslash |u\; +\; v\backslash |^2\; -\; \backslash |u\; -\; v\backslash |^2\backslash bigr)\backslash ,.$$

For complex Hilbert spaces, it is

$$\backslash langle\; u,\; v\backslash rangle\; =\; \backslash tfrac\{1\}\{4\}\backslash bigl(\backslash |u\; +\; v\backslash |^2\; -\; \backslash |u\; -\; v\backslash |^2\; +\; i\backslash |u\; +\; iv\backslash |^2\; -\; i\backslash |u\; -\; iv\backslash |^2\backslash bigr)\backslash ,.$$

The parallelogram law implies that any Hilbert space is a uniformly convex Banach space.{{harvnb|Clarkson|1936}}

This subsection employs the Hilbert projection theorem. If {{math|C}} is a non-empty closed convex subset of a Hilbert space {{math|H}} and {{math|x}} a point in {{math|H}}, there exists a unique point {{math|y ∈ C}} that minimizes the distance between {{math|x}} and points in {{math|C}},{{harvnb|Rudin|1987|loc=Theorem 4.10}}

$$y\; \backslash in\; C\; \backslash ,,\; \backslash quad\; \backslash |x\; -\; y\backslash |\; =\; \backslash operatorname\{dist\}(x,\; C)\; =\; \backslash min\; \backslash bigl\backslash \{\; \backslash |x\; -\; z\backslash |\; \backslash mathrel\{\backslash big|\}\; z\; \backslash in\; C\; \backslash bigr\backslash \}\backslash ,.$$

This is equivalent to saying that there is a point with minimal norm in the translated convex set {{math|1=D = C − x}}. The proof consists in showing that every minimizing sequence {{math|(d_n) ⊂ D}} is Cauchy (using the parallelogram identity) hence converges (using completeness) to a point in {{math|D}} that has minimal norm. More generally, this holds in any uniformly convex Banach space.{{harvnb|Dunford|Schwartz|1958|loc=II.4.29}}

When this result is applied to a closed subspace {{math|F}} of {{math|H}}, it can be shown that the point {{math|y ∈ F}} closest to {{math|x}} is characterized by{{harvnb|Rudin|1987|loc=Theorem 4.11}}

$$y\; \backslash in\; F\; \backslash ,,\; \backslash quad\; x\; -\; y\; \backslash perp\; F\; \backslash ,.$$

This point {{math|y}} is the orthogonal projection of {{math|x}} onto {{math|F}}, and the mapping {{math|P_F : x → y}} is linear (see {{slink|#Orthogonal complements and projections}}). This result is especially significant in applied mathematics, especially numerical analysis, where it forms the basis of least squares methods.

{{cite book

| last1 = Blanchet | first1 = Gérard | last2 = Charbit | first2 = Maurice

| title = Digital Signal and Image Processing Using MATLAB

| publisher = Wiley | volume = 1 | edition = Second | date = 2014

| location = New Jersey | pages = 349–360 | isbn = 978-1848216402

}}

In particular, when {{math|F}} is not equal to {{math|H}}, one can find a nonzero vector {{math|v}} orthogonal to {{math|F}} (select {{math|x ∉ F}} and {{math|1=v = x − y}}). A very useful criterion is obtained by applying this observation to the closed subspace {{math|F}} generated by a subset {{math|S}} of {{math|H}}.

: A subset {{math|S}} of {{math|H}} spans a dense vector subspace if (and only if) the vector 0 is the sole vector {{math|v ∈ H}} orthogonal to {{math|S}}.

The dual space {{math|H*}} is the space of all continuous linear functions from the space {{math|H}} into the base field. It carries a natural norm, defined by

$$\backslash |\backslash varphi\backslash |\; =\; \backslash sup\_\{\backslash |x\backslash |=1,\; x\backslash in\; H\}\; |\backslash varphi(x)|\; \backslash ,.$$

This norm satisfies the parallelogram law, and so the dual space is also an inner product space where this inner product can be defined in terms of this dual norm by using the polarization identity. The dual space is also complete so it is a Hilbert space in its own right.

If {{math|1=e_• = (e_i)_{i ∈ I}}} is a complete orthonormal basis for {{mvar|H}} then the inner product on the dual space of any two $f,\; g\; \backslash in\; H^*$ is

$$\backslash langle\; f,\; g\; \backslash rangle\_\{H^\{*\}\}\; =\; \backslash sum\_\{i\; \backslash in\; I\}\; f\; (e\_i)\; \backslash overline\{g\; (e\_i)\}$$

where all but countably many of the terms in this series are zero.

The Riesz representation theorem affords a convenient description of the dual space. To every element {{math|u}} of {{math|H}}, there is a unique element {{math|φ_u}} of {{math|H*}}, defined by

$$\backslash varphi\_u(x)\; =\; \backslash langle\; x,\; u\backslash rangle$$

where moreover, $\backslash left\backslash |\; \backslash varphi\_u\; \backslash right\backslash |\; =\; \backslash left\backslash |\; u\; \backslash right\backslash |.$

The Riesz representation theorem states that the map from {{math|H}} to {{math|H*}} defined by {{math|u ↦ φ_u}} is surjective, which makes this map an isometric antilinear isomorphism.{{harvnb|Weidmann|1980|loc=Theorem 4.8}} So to every element {{math|φ}} of the dual {{math|H*}} there exists one and only one {{math|u_φ}} in {{math|H}} such that

$$\backslash langle\; x,\; u\_\backslash varphi\backslash rangle\; =\; \backslash varphi(x)$$

for all {{math|x ∈ H}}. The inner product on the dual space {{math|H*}} satisfies

$$\backslash langle\; \backslash varphi,\; \backslash psi\; \backslash rangle\; =\; \backslash langle\; u\_\backslash psi,\; u\_\backslash varphi\; \backslash rangle\; \backslash ,.$$

The reversal of order on the right-hand side restores linearity in {{math|φ}} from the antilinearity of {{math|u_φ}}. In the real case, the antilinear isomorphism from {{math|H}} to its dual is actually an isomorphism, and so real Hilbert spaces are naturally isomorphic to their own duals.

The representing vector {{math|u_φ}} is obtained in the following way. When {{math|φ ≠ 0}}, the kernel {{math|1=F = Ker(φ)}} is a closed vector subspace of {{math|H}}, not equal to {{math|H}}, hence there exists a nonzero vector {{math|v}} orthogonal to {{math|F}}. The vector {{math|u}} is a suitable scalar multiple {{math|λv}} of {{math|v}}. The requirement that {{math|1=φ(v) = ⟨v, u⟩}} yields

$$u\; =\; \backslash langle\; v,\; v\; \backslash rangle^\{-1\}\; \backslash ,\; \backslash overline\{\backslash varphi\; (v)\}\; \backslash ,\; v\; \backslash ,.$$

This correspondence {{math|φ ↔ u}} is exploited by the bra–ket notation popular in physics.{{harvnb|Peres|1993|pp=77–78}} It is common in physics to assume that the inner product, denoted by {{math|{{bra-ket|x|y}}}}, is linear on the right,

$$\backslash langle\; x\; |\; y\; \backslash rangle\; =\; \backslash langle\; y,\; x\; \backslash rangle\; \backslash ,.$$

The result {{math|{{bra-ket|x|y}}}} can be seen as the action of the linear functional {{math|{{bra|x}}}} (the bra) on the vector {{math|{{ket|y}}}} (the ket).

The Riesz representation theorem relies fundamentally not just on the presence of an inner product, but also on the completeness of the space. In fact, the theorem implies that the topological dual of any inner product space can be identified with its completion.{{harvtxt|Weidmann|1980|loc=Exercise 4.11}} An immediate consequence of the Riesz representation theorem is also that a Hilbert space {{math|H}} is reflexive, meaning that the natural map from {{math|H}} into its double dual space is an isomorphism.

{{main|Weak convergence (Hilbert space)}}

In a Hilbert space {{math|H}}, a sequence {{math|{x_n}}} is weakly convergent to a vector {{math|x ∈ H}} when

$$\backslash lim\_n\; \backslash langle\; x\_n,\; v\; \backslash rangle\; =\; \backslash langle\; x,\; v\; \backslash rangle$$

for every {{math|v ∈ H}}.

For example, any orthonormal sequence {{math|{f_n}}} converges weakly to 0, as a consequence of Bessel's inequality. Every weakly convergent sequence {{math|{x_n}}} is bounded, by the uniform boundedness principle.

Conversely, every bounded sequence in a Hilbert space admits weakly convergent subsequences (Alaoglu's theorem).{{harvnb|Weidmann|1980|loc=§4.5}} This fact may be used to prove minimization results for continuous convex functionals, in the same way that the Bolzano–Weierstrass theorem is used for continuous functions on {{math|R^d}}. Among several variants, one simple statement is as follows:{{harvnb|Buttazzo|Giaquinta|Hildebrandt|1998|loc=Theorem 5.17}}

: If {{math|f : H → R}} is a convex continuous function such that {{math|f(x)}} tends to {{math|+∞}} when {{math|{{norm|x}}}} tends to {{math|∞}}, then {{math|f}} admits a minimum at some point {{math|x₀ ∈ H}}.

This fact (and its various generalizations) are fundamental for direct methods in the calculus of variations. Minimization results for convex functionals are also a direct consequence of the slightly more abstract fact that closed bounded convex subsets in a Hilbert space {{math|H}} are weakly compact, since {{math|H}} is reflexive. The existence of weakly convergent subsequences is a special case of the Eberlein–Šmulian theorem.

Any general property of Banach spaces continues to hold for Hilbert spaces. The open mapping theorem states that a continuous surjective linear transformation from one Banach space to another is an open mapping meaning that it sends open sets to open sets. A corollary is the bounded inverse theorem, that a continuous and bijective linear function from one Banach space to another is an isomorphism (that is, a continuous linear map whose inverse is also continuous). This theorem is considerably simpler to prove in the case of Hilbert spaces than in general Banach spaces.{{harvnb|Halmos|1982|loc=Problem 52, 58}} The open mapping theorem is equivalent to the closed graph theorem, which asserts that a linear function from one Banach space to another is continuous if and only if its graph is a closed set.{{harvnb|Rudin|1973}} In the case of Hilbert spaces, this is basic in the study of unbounded operators (see Closed operator).

The (geometrical) Hahn–Banach theorem asserts that a closed convex set can be separated from any point outside it by means of a hyperplane of the Hilbert space. This is an immediate consequence of the best approximation property: if {{math|y}} is the element of a closed convex set {{math|F}} closest to {{math|x}}, then the separating hyperplane is the plane perpendicular to the segment {{math|xy}} passing through its midpoint.{{harvnb|Trèves|1967|loc=Chapter 18}}

The continuous linear operators {{math|A : H₁ → H₂}} from a Hilbert space {{math|H₁}} to a second Hilbert space {{math|H₂}} are bounded in the sense that they map bounded sets to bounded sets.A general reference for this section is {{harvtxt|Rudin|1973}}, chapter 12. Conversely, if an operator is bounded, then it is continuous. The space of such bounded linear operators has a norm, the operator norm given by

$$\backslash lVert\; A\; \backslash rVert\; =\; \backslash sup\; \backslash bigl\backslash \{\backslash |\; Ax\; \backslash |\; \backslash mathrel\{\backslash big|\}\; \backslash |\; x\; \backslash |\; \backslash leq\; 1\; \backslash bigr\backslash \}\backslash ,.$$

The sum and the composite of two bounded linear operators is again bounded and linear. For y in H₂, the map that sends {{math|x ∈ H₁}} to {{math|⟨Ax, y⟩}} is linear and continuous, and according to the Riesz representation theorem can therefore be represented in the form

$$\backslash left\backslash langle\; x,\; A^*\; y\; \backslash right\backslash rangle\; =\; \backslash langle\; Ax,\; y\; \backslash rangle$$

for some vector {{math|A*y}} in {{math|H₁}}. This defines another bounded linear operator {{math|A* : H₂ → H₁}}, the adjoint of {{mvar|A}}. The adjoint satisfies {{math|1=A** = A}}. When the Riesz representation theorem is used to identify each Hilbert space with its continuous dual space, the adjoint of {{mvar|A}} can be shown to be identical to the transpose {{math|^tA : H₂* → H₁*}} of {{mvar|A}}, which by definition sends $\backslash psi\; \backslash in\; H\_2^\{*\}$ to the functional $\backslash psi\; \backslash circ\; A\; \backslash in\; H\_1^\{*\}.$

The set {{math|B(H)}} of all bounded linear operators on {{math|H}} (meaning operators {{math|H → H}}), together with the addition and composition operations, the norm and the adjoint operation, is a C*-algebra, which is a type of operator algebra.

An element {{math|A}} of {{math|B(H)}} is called 'self-adjoint' or 'Hermitian' if {{math|1=A* = A}}. If {{math|A}} is Hermitian and {{math|⟨Ax, x⟩ ≥ 0}} for every {{math|x}}, then {{math|A}} is called 'nonnegative', written {{math|A ≥ 0}}; if equality holds only when {{math|1=x = 0}}, then {{math|A}} is called 'positive'. The set of self adjoint operators admits a partial order, in which {{math|A ≥ B}} if {{math|A − B ≥ 0}}. If {{math|A}} has the form {{math|B*B}} for some {{math|B}}, then {{math|A}} is nonnegative; if {{math|B}} is invertible, then {{math|A}} is positive. A converse is also true in the sense that, for a non-negative operator {{math|A}}, there exists a unique non-negative square root {{math|B}} such that

$$A\; =\; B^2\; =\; B^*B\backslash ,.$$

In a sense made precise by the spectral theorem, self-adjoint operators can usefully be thought of as operators that are "real". An element {{math|A}} of {{math|B(H)}} is called normal if {{math|1=A*A = AA*}}. Normal operators decompose into the sum of a self-adjoint operator and an imaginary multiple of a self adjoint operator

$$A\; =\; \backslash frac\{A\; +\; A^*\}\{2\}\; +\; i\backslash frac\{A\; -\; A^*\}\{2i\}$$

that commute with each other. Normal operators can also usefully be thought of in terms of their real and imaginary parts.

An element {{math|U}} of {{math|B(H)}} is called unitary if {{math|U}} is invertible and its inverse is given by {{math|U*}}. This can also be expressed by requiring that {{math|U}} be onto and {{math|1=⟨Ux, Uy⟩ = ⟨x, y⟩}} for all {{math|x, y ∈ H}}. The unitary operators form a group under composition, which is the isometry group of {{math|H}}.

An element of {{math|B(H)}} is compact if it sends bounded sets to relatively compact sets. Equivalently, a bounded operator {{math|T}} is compact if, for any bounded sequence {{math|{x_k}}}, the sequence {{math|{Tx_k}}} has a convergent subsequence. Many integral operators are compact, and in fact define a special class of operators known as Hilbert–Schmidt operators that are especially important in the study of integral equations. Fredholm operators differ from a compact operator by a multiple of the identity, and are equivalently characterized as operators with a finite dimensional kernel and cokernel. The index of a Fredholm operator {{math|T}} is defined by

$$\backslash operatorname\{index\}\; T\; =\; \backslash dim\backslash ker\; T\; -\; \backslash dim\backslash operatorname\{coker\}\; T\; \backslash ,.$$

The index is homotopy invariant, and plays a deep role in differential geometry via the Atiyah–Singer index theorem.

Unbounded operators are also tractable in Hilbert spaces, and have important applications to quantum mechanics.See {{harvtxt|Prugovečki|1981}}, {{harvtxt|Reed|Simon|1980|loc=Chapter VIII}} and {{harvtxt|Folland|1989}}. An unbounded operator {{math|T}} on a Hilbert space {{math|H}} is defined as a linear operator whose domain {{math|D(T)}} is a linear subspace of {{math|H}}. Often the domain {{math|D(T)}} is a dense subspace of {{math|H}}, in which case {{math|T}} is known as a densely defined operator.

The adjoint of a densely defined unbounded operator is defined in essentially the same manner as for bounded operators. Self-adjoint unbounded operators play the role of the observables in the mathematical formulation of quantum mechanics. Examples of self-adjoint unbounded operators on the Hilbert space {{math|L²(R)}} are:{{harvnb|Prugovečki|1981|loc=III, §1.4}}

• A suitable extension of the differential operator $$(A\; f)(x)\; =\; -i\; \backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}x\}\; f(x)\; \backslash ,,$$ where {{math|i}} is the imaginary unit and {{math|f}} is a differentiable function of compact support.
• The multiplication-by-{{math|x}} operator: $$(B\; f)\; (x)\; =\; x\; f(x)\backslash ,.$$

These correspond to the momentum and position observables, respectively. Neither {{math|A}} nor {{math|B}} is defined on all of {{math|H}}, since in the case of {{math|A}} the derivative need not exist, and in the case of {{math|B}} the product function need not be square integrable. In both cases, the set of possible arguments form dense subspaces of {{math|L²(R)}}.

Two Hilbert spaces {{math|H₁}} and {{math|H₂}} can be combined into another Hilbert space, called the (orthogonal) direct sum,{{harvnb|Dunford|Schwartz|1958|loc=IV.4.17-18}} and denoted

$$H\_1\; \backslash oplus\; H\_2\; \backslash ,,$$

consisting of the set of all ordered pairs {{math|(x₁, x₂)}} where {{math|x_i ∈ H_i}}, {{math|1=i = 1, 2}}, and inner product defined by

$$\backslash bigl\backslash langle\; (x\_1,\; x\_2),\; (y\_1,\; y\_2)\backslash bigr\backslash rangle\_\{H\_1\; \backslash oplus\; H\_2\}\; =\; \backslash left\backslash langle\; x\_1,\; y\_1\backslash right\backslash rangle\_\{H\_1\}\; +\; \backslash left\backslash langle\; x\_2,\; y\_2\backslash right\backslash rangle\_\{H\_2\}\; \backslash ,.$$

More generally, if {{math|H_i}} is a family of Hilbert spaces indexed by {{nowrap|i ∈ I}}, then the direct sum of the {{math|H_i}}, denoted

$$\backslash bigoplus\_\{i\; \backslash in\; I\}H\_i$$

consists of the set of all indexed families

$$x\; =\; (x\_i\; \backslash in\; H\_i\; \backslash mid\; i\; \backslash in\; I)\; \backslash in\; \backslash prod\_\{i\; \backslash in\; I\}H\_i$$

in the Cartesian product of the {{math|H_i}} such that

The inner product is defined by

$$\backslash langle\; x,\; y\backslash rangle\; =\; \backslash sum\_\{i\; \backslash in\; I\}\; \backslash left\backslash langle\; x\_i,\; y\_i\backslash right\backslash rangle\_\{H\_i\}\; \backslash ,.$$

Each of the {{math|H_i}} is included as a closed subspace in the direct sum of all of the {{math|H_i}}. Moreover, the {{math|H_i}} are pairwise orthogonal. Conversely, if there is a system of closed subspaces, {{math|V_i}}, {{math|i ∈ I}}, in a Hilbert space {{math|H}}, that are pairwise orthogonal and whose union is dense in {{math|H}}, then {{math|H}} is canonically isomorphic to the direct sum of {{math|V_i}}. In this case, {{math|H}} is called the internal direct sum of the {{math|V_i}}. A direct sum (internal or external) is also equipped with a family of orthogonal projections {{math|E_i}} onto the {{math|i}}th direct summand {{math|H_i}}. These projections are bounded, self-adjoint, idempotent operators that satisfy the orthogonality condition

$$E\_i\; E\_j\; =\; 0,\backslash quad\; i\; \backslash neq\; j\; \backslash ,.$$

The spectral theorem for compact self-adjoint operators on a Hilbert space {{math|H}} states that {{math|H}} splits into an orthogonal direct sum of the eigenspaces of an operator, and also gives an explicit decomposition of the operator as a sum of projections onto the eigenspaces. The direct sum of Hilbert spaces also appears in quantum mechanics as the Fock space of a system containing a variable number of particles, where each Hilbert space in the direct sum corresponds to an additional degree of freedom for the quantum mechanical system. In representation theory, the Peter–Weyl theorem guarantees that any unitary representation of a compact group on a Hilbert space splits as the direct sum of finite-dimensional representations.

{{main|Tensor product of Hilbert spaces}}

If {{math|x₁, y₁ ∊ H₁}} and {{math|x₂, y₂ ∊ H₂}}, then one defines an inner product on the (ordinary) tensor product as follows. On simple tensors, let

$$\backslash langle\; x\_1\; \backslash otimes\; x\_2,\; \backslash ,\; y\_1\; \backslash otimes\; y\_2\; \backslash rangle\; =\; \backslash langle\; x\_1,\; y\_1\; \backslash rangle\; \backslash ,\; \backslash langle\; x\_2,\; y\_2\; \backslash rangle\; \backslash ,.$$

This formula then extends by sesquilinearity to an inner product on {{math|H₁ ⊗ H₂}}. The Hilbertian tensor product of {{math|H₁}} and {{math|H₂}}, sometimes denoted by {{math|H₁ $\backslash widehat\{\backslash otimes\}$ H₂}}, is the Hilbert space obtained by completing {{math|H₁ ⊗ H₂}} for the metric associated to this inner product.{{harvnb|Weidmann|1980|loc=§3.4}}

An example is provided by the Hilbert space {{math|L²([0, 1])}}. The Hilbertian tensor product of two copies of {{math|L²([0, 1])}} is isometrically and linearly isomorphic to the space {{math|L²([0, 1]²)}} of square-integrable functions on the square {{math|[0, 1]²}}. This isomorphism sends a simple tensor {{math|f₁ ⊗ f₂}} to the function

$$(s,\; t)\; \backslash mapsto\; f\_1(s)\; \backslash ,\; f\_2(t)$$

on the square.

This example is typical in the following sense.{{harvnb|Kadison|Ringrose|1983|loc=Theorem 2.6.4}} Associated to every simple tensor product {{math|x₁ ⊗ x₂}} is the rank one operator from {{math|H{{su|p=∗|b=1|lh=0.8em}}}} to {{math|H₂}} that maps a given {{math|x* ∈ H{{su|p=∗|b=1|lh=0.8em}}}} as

$$x^*\; \backslash mapsto\; x^*(x\_1)\; x\_2\; \backslash ,.$$

This mapping defined on simple tensors extends to a linear identification between {{math|H₁ ⊗ H₂}} and the space of finite rank operators from {{math|H{{su|p=∗|b=1|lh=0.8em}}}} to {{math|H₂}}. This extends to a linear isometry of the Hilbertian tensor product {{math|H₁ $\backslash widehat\{\backslash otimes\}$ H₂}} with the Hilbert space {{math|HS(H{{su|p=∗|b=1|lh=0.8em}}, H₂)}} of Hilbert–Schmidt operators from {{math|H{{su|p=∗|b=1|lh=0.8em}}}} to {{math|H₂}}.

The notion of an orthonormal basis from linear algebra generalizes over to the case of Hilbert spaces.{{harvnb|Dunford|Schwartz|1958|loc=§IV.4}} In a Hilbert space {{math|H}}, an orthonormal basis is a family {{math|{e_k}_{k ∈ B}}} of elements of {{math|H}} satisfying the conditions:

1. Orthogonality: Every two different elements of {{math|B}} are orthogonal: {{math|1=⟨e_k, e_j⟩ = 0}} for all {{math|k, j ∈ B}} with {{nowrap|k ≠ j}}.
2. Normalization: Every element of the family has norm 1: {{math|1={{norm|e_k}} = 1}} for all {{math|k ∈ B}}.
3. Completeness: The linear span of the family {{math|e_k}}, {{math|k ∈ B}}, is dense in H.

A system of vectors satisfying the first two conditions basis is called an orthonormal system or an orthonormal set (or an orthonormal sequence if {{math|B}} is countable). Such a system is always linearly independent.

Despite the name, an orthonormal basis is not, in general, a basis in the sense of linear algebra (Hamel basis). More precisely, an orthonormal basis is a Hamel basis if and only if the Hilbert space is a finite-dimensional vector space.{{harvnb|Roman|2008|loc=p. 218}}

Completeness of an orthonormal system of vectors of a Hilbert space can be equivalently restated as:

: for every {{math|v ∈ H}}, if {{math|1=⟨v, e_k⟩ = 0}} for all {{math|k ∈ B}}, then {{math|1=v = 0}}.

This is related to the fact that the only vector orthogonal to a dense linear subspace is the zero vector, for if {{math|S}} is any orthonormal set and {{math|v}} is orthogonal to {{math|S}}, then {{math|v}} is orthogonal to the closure of the linear span of {{math|S}}, which is the whole space.

Examples of orthonormal bases include:

• the set {{math|{(1, 0, 0), (0, 1, 0), (0, 0, 1)}}} forms an orthonormal basis of {{math|R³}} with the dot product;
• the sequence {{math|{ f_n {{!}} n ∈ Z}}} with {{math|1=f_n(x) = exp(2πinx)}} forms an orthonormal basis of the complex space {{math|L²([0, 1])}};

In the infinite-dimensional case, an orthonormal basis will not be a basis in the sense of linear algebra; to distinguish the two, the latter basis is also called a Hamel basis. That the span of the basis vectors is dense implies that every vector in the space can be written as the sum of an infinite series, and the orthogonality implies that this decomposition is unique.

The space $\backslash ell\_2$ of square-summable sequences of complex numbers is the set of infinite sequences

$$(c\_1,\; c\_2,\; c\_3,\; \backslash dots)$$

of real or complex numbers such that

This space has an orthonormal basis:

$$\backslash begin\{align\}$$

e_1 &= (1, 0, 0, \dots) \\

e_2 &= (0, 1, 0, \dots) \\

& \ \ \vdots

\end{align}

This space is the infinite-dimensional generalization of the $\backslash ell\_2^n$ space of finite-dimensional vectors. It is usually the first example used to show that in infinite-dimensional spaces, a set that is closed and bounded is not necessarily (sequentially) compact (as is the case in all finite dimensional spaces). Indeed, the set of orthonormal vectors above shows this: It is an infinite sequence of vectors in the unit ball (i.e., the ball of points with norm less than or equal one). This set is clearly bounded and closed; yet, no subsequence of these vectors converges to anything and consequently the unit ball in $\backslash ell\_2$ is not compact. Intuitively, this is because "there is always another coordinate direction" into which the next elements of the sequence can evade.

One can generalize the space $\backslash ell\_2$ in many ways. For example, if {{math|B}} is any set, then one can form a Hilbert space of sequences with index set {{math|B}}, defined by{{harvnb|Rudin|1987}}, Definition 3.7

The summation over B is here defined by

$$\backslash sum\_\{b\; \backslash in\; B\}\; \backslash left|x\; (b)\backslash right|^2\; =\; \backslash sup\; \backslash sum\_\{n=1\}^N\; \backslash left|x(b\_n)\backslash right|^2$$

the supremum being taken over all finite subsets of {{math|B}}. It follows that, for this sum to be finite, every element of {{math|l²(B)}} has only countably many nonzero terms. This space becomes a Hilbert space with the inner product

$$\backslash langle\; x,\; y\; \backslash rangle\; =\; \backslash sum\_\{b\; \backslash in\; B\}\; x(b)\backslash overline\{y(b)\}$$

for all {{math|x, y ∈ l²(B)}}. Here the sum also has only countably many nonzero terms, and is unconditionally convergent by the Cauchy–Schwarz inequality.

An orthonormal basis of {{math|l²(B)}} is indexed by the set {{math|B}}, given by

: $$e\_b(b\text{'})\; =\; \backslash begin\{cases\}$$

1 & \text{if } b=b'\\

0 & \text{otherwise.}

\end{cases}

{{Anchor|Bessel's inequality}}

{{Anchor|Parseval's formula}}

Let {{math|f₁, ..., f_n}} be a finite orthonormal system in {{math|H}}. For an arbitrary vector {{math|x ∈ H}}, let

$$y\; =\; \backslash sum\_\{j=1\}^n\; \backslash langle\; x,\; f\_j\; \backslash rangle\; \backslash ,\; f\_j\; \backslash ,.$$

Then {{math|1=⟨x, f_k⟩ = ⟨y, f_k⟩}} for every {{math|1=k = 1, ..., n}}. It follows that {{math|x − y}} is orthogonal to each {{math|f_k}}, hence {{math|x − y}} is orthogonal to {{math|y}}. Using the Pythagorean identity twice, it follows that

$$\backslash |x\backslash |^2\; =\; \backslash |x\; -\; y\backslash |^2\; +\; \backslash |y\backslash |^2\; \backslash ge\; \backslash |y\backslash |^2\; =\; \backslash sum\_\{j=1\}^n\backslash bigl|\backslash langle\; x,\; f\_j\; \backslash rangle\backslash bigr|^2\; \backslash ,.$$

Let {{math|{f_i}, i ∈ I}}, be an arbitrary orthonormal system in {{math|H}}. Applying the preceding inequality to every finite subset {{math|J}} of {{math|I}} gives Bessel's inequality:For the case of finite index sets, see, for instance, {{harvnb|Halmos|1957|loc=§5}}. For infinite index sets, see {{harvnb|Weidmann|1980|loc=Theorem 3.6}}.

$$\backslash sum\_\{i\; \backslash in\; I\}\backslash bigl|\backslash langle\; x,\; f\_i\; \backslash rangle\backslash bigr|^2\; \backslash le\; \backslash |x\backslash |^2,\; \backslash quad\; x\; \backslash in\; H$$

(according to the definition of the sum of an arbitrary family of non-negative real numbers).

Geometrically, Bessel's inequality implies that the orthogonal projection of {{math|x}} onto the linear subspace spanned by the {{math|f_i}} has norm that does not exceed that of {{math|x}}. In two dimensions, this is the assertion that the length of the leg of a right triangle may not exceed the length of the hypotenuse.

Bessel's inequality is a stepping stone to the stronger result called Parseval's identity, which governs the case when Bessel's inequality is actually an equality. By definition, if {{math|{e_k}_{k ∈ B}}} is an orthonormal basis of {{math|H}}, then every element {{math|x}} of {{math|H}} may be written as

$$x\; =\; \backslash sum\_\{k\; \backslash in\; B\}\; \backslash left\backslash langle\; x,\; e\_k\; \backslash right\backslash rangle\; \backslash ,\; e\_k\; \backslash ,.$$

Even if {{math|B}} is uncountable, Bessel's inequality guarantees that the expression is well-defined and consists only of countably many nonzero terms. This sum is called the Fourier expansion of {{math|x}}, and the individual coefficients {{math|⟨x, e_k⟩}} are the Fourier coefficients of {{math|x}}. Parseval's identity then asserts that{{harvtxt|Hewitt|Stromberg|1965|loc=Theorem 16.26}}

$$\backslash |x\backslash |^2\; =\; \backslash sum\_\{k\backslash in\; B\}|\backslash langle\; x,\; e\_k\backslash rangle|^2\; \backslash ,.$$

Conversely, if {{math|{e_k}}} is an orthonormal set such that Parseval's identity holds for every {{math|x}}, then {{math|{e_k}}} is an orthonormal basis.

As a consequence of Zorn's lemma, every Hilbert space admits an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same cardinality, called the Hilbert dimension of the space.{{harvnb|Levitan|2001}}. Many authors, such as {{harvtxt|Dunford|Schwartz|1958|loc=§IV.4}}, refer to this just as the dimension. Unless the Hilbert space is finite dimensional, this is not the same thing as its dimension as a linear space (the cardinality of a Hamel basis). For instance, since {{math|l²(B)}} has an orthonormal basis indexed by {{math|B}}, its Hilbert dimension is the cardinality of {{math|B}} (which may be a finite integer, or a countable or uncountable cardinal number).

The Hilbert dimension is not greater than the Hamel dimension (the usual dimension of a vector space).

As a consequence of Parseval's identity,{{harvtxt|Hewitt|Stromberg|1965|loc=Theorem 16.29}} if {{math|{e_k}_{k ∈ B}}} is an orthonormal basis of {{math|H}}, then the map {{math|Φ : H → l²(B)}} defined by {{math|1=Φ(x) = ⟨x, e_k⟩_k∈B}} is an isometric isomorphism of Hilbert spaces: it is a bijective linear mapping such that

$$\backslash bigl\backslash langle\; \backslash Phi\; (x),\; \backslash Phi(y)\; \backslash bigr\backslash rangle\_\{l^2(B)\}\; =\; \backslash left\backslash langle\; x,\; y\; \backslash right\backslash rangle\_H$$

for all {{math|x, y ∈ H}}. The cardinal number of {{math|B}} is the Hilbert dimension of {{math|H}}. Thus every Hilbert space is isometrically isomorphic to a sequence space {{math|l²(B)}} for some set {{math|B}}.

By definition, a Hilbert space is separable provided it contains a dense countable subset. Along with Zorn's lemma, this means a Hilbert space is separable if and only if it admits a countable orthonormal basis. All infinite-dimensional separable Hilbert spaces are therefore isometrically isomorphic to the square-summable sequence space $\backslash ell^2.$

In the past, Hilbert spaces were often required to be separable as part of the definition.{{harvnb|Prugovečki|1981|loc=I, §4.2}}

Most spaces used in physics are separable, and since these are all isomorphic to each other, one often refers to any infinite-dimensional separable Hilbert space as "the Hilbert space" or just "Hilbert space".{{harvtxt|von Neumann|1955}} defines a Hilbert space via a countable Hilbert basis, which amounts to an isometric isomorphism with l². The convention still persists in most rigorous treatments of quantum mechanics; see for instance {{harvnb|Sobrino|1996|loc=Appendix B}}. Even in quantum field theory, most of the Hilbert spaces are in fact separable, as stipulated by the Wightman axioms. However, it is sometimes argued that non-separable Hilbert spaces are also important in quantum field theory, roughly because the systems in the theory possess an infinite number of degrees of freedom and any infinite Hilbert tensor product (of spaces of dimension greater than one) is non-separable.{{harvnb|Streater|Wightman|1964|pp=86–87}} For instance, a bosonic field can be naturally thought of as an element of a tensor product whose factors represent harmonic oscillators at each point of space. From this perspective, the natural state space of a boson might seem to be a non-separable space. However, it is only a small separable subspace of the full tensor product that can contain physically meaningful fields (on which the observables can be defined). Another non-separable Hilbert space models the state of an infinite collection of particles in an unbounded region of space. An orthonormal basis of the space is indexed by the density of the particles, a continuous parameter, and since the set of possible densities is uncountable, the basis is not countable.

{{Main|Orthogonal complement|}}

If {{math|S}} is a subset of a Hilbert space {{math|H}}, the set of vectors orthogonal to {{math|S}} is defined by

$$S^\backslash perp\; =\; \backslash left\backslash \{\; x\; \backslash in\; H\; \backslash mid\; \backslash langle\; x,\; s\; \backslash rangle\; =\; 0\backslash \; \backslash text\{\; for\; all\; \}\; s\; \backslash in\; S\; \backslash right\backslash \}\; \backslash ,.$$

The set {{math|S^⊥}} is a closed subspace of {{math|H}} (can be proved easily using the linearity and continuity of the inner product) and so forms itself a Hilbert space. If {{math|V}} is a closed subspace of {{math|H}}, then {{math|V^⊥}} is called the {{em|orthogonal complement}} of {{math|V}}. In fact, every {{math|x ∈ H}} can then be written uniquely as {{math|1=x = v + w}}, with {{math|v ∈ V}} and {{math|w ∈ V^⊥}}. Therefore, {{math|H}} is the internal Hilbert direct sum of {{math|V}} and {{math|V^⊥}}.

The linear operator {{math|P_V : H → H}} that maps {{math|x}} to {{math|v}} is called the {{em|orthogonal projection}} onto {{math|V}}. There is a natural one-to-one correspondence between the set of all closed subspaces of {{math|H}} and the set of all bounded self-adjoint operators {{math|P}} such that {{math|1=P² = P}}. Specifically,

{{math theorem|The orthogonal projection {{math|P_V}} is a self-adjoint linear operator on {{math|H}} of norm ≤ 1 with the property {{math|1=P{{su|p=2|b=V}} = P_V}}. Moreover, any self-adjoint linear operator {{math|E}} such that {{math|1=E² = E}} is of the form {{math|P_V}}, where {{math|V}} is the range of {{math|E}}. For every {{math|x}} in {{math|H}}, {{math|P_V(x)}} is the unique element {{math|v}} of {{math|V}} that minimizes the distance {{math|{{norm|x − v}}}}.}}

This provides the geometrical interpretation of {{math|P_V(x)}}: it is the best approximation to x by elements of V.{{harvnb|Young|1988|loc=Theorem 15.3}}

Projections {{math|P_U}} and {{math|P_V}} are called mutually orthogonal if {{math|1=P_UP_V = 0}}. This is equivalent to {{math|U}} and {{math|V}} being orthogonal as subspaces of {{math|H}}. The sum of the two projections {{math|P_U}} and {{math|P_V}} is a projection only if {{math|U}} and {{math|V}} are orthogonal to each other, and in that case {{math|1=P_U + P_V = P_U+V}}.{{harvnb|von Neumann|1955}}, Theorem 16 The composite {{math|P_UP_V}} is generally not a projection; in fact, the composite is a projection if and only if the two projections commute, and in that case {{math|1=P_UP_V = P_U∩V}}.{{harvnb|von Neumann|1955|loc=Theorem 14}}

By restricting the codomain to the Hilbert space {{math|V}}, the orthogonal projection {{math|P_V}} gives rise to a projection mapping {{math|π : H → V}}; it is the adjoint of the inclusion mapping

$$i\; :\; V\; \backslash to\; H\; \backslash ,,$$

meaning that

$$\backslash left\backslash langle\; i\; x,\; y\backslash right\backslash rangle\_H\; =\; \backslash left\backslash langle\; x,\; \backslash pi\; y\; \backslash right\backslash rangle\_V$$

for all {{math|x ∈ V}} and {{math|y ∈ H}}.

The operator norm of the orthogonal projection {{math|P_V}} onto a nonzero closed subspace {{math|V}} is equal to 1:

$$\backslash |P\_V\backslash |\; =\; \backslash sup\_\{x\; \backslash in\; H,\; x\; \backslash neq\; 0\}\; \backslash frac\{\backslash |P\_V\; x\backslash |\}\{\backslash |x\backslash |\}\; =\; 1\; \backslash ,.$$

Every closed subspace V of a Hilbert space is therefore the image of an operator {{math|P}} of norm one such that {{math|1=P² = P}}. The property of possessing appropriate projection operators characterizes Hilbert spaces:{{harvnb|Kakutani|1939}}

• A Banach space of dimension higher than 2 is (isometrically) a Hilbert space if and only if, for every closed subspace {{math|V}}, there is an operator {{math|P_V}} of norm one whose image is {{math|V}} such that {{math|1=P{{su|b=V|p=2}} = P_V}}.

While this result characterizes the metric structure of a Hilbert space, the structure of a Hilbert space as a topological vector space can itself be characterized in terms of the presence of complementary subspaces:{{harvnb|Lindenstrauss|Tzafriri|1971}}

• A Banach space {{math|X}} is topologically and linearly isomorphic to a Hilbert space if and only if, to every closed subspace {{math|V}}, there is a closed subspace {{math|W}} such that {{math|X}} is equal to the internal direct sum {{math|V ⊕ W}}.

The orthogonal complement satisfies some more elementary results. It is a monotone function in the sense that if {{math|U ⊂ V}}, then {{math|V^⊥ ⊆ U^⊥}} with equality holding if and only if {{math|V}} is contained in the closure of {{math|U}}. This result is a special case of the Hahn–Banach theorem. The closure of a subspace can be completely characterized in terms of the orthogonal complement: if {{math|V}} is a subspace of {{math|H}}, then the closure of {{math|V}} is equal to {{math|V^⊥⊥}}. The orthogonal complement is thus a Galois connection on the partial order of subspaces of a Hilbert space. In general, the orthogonal complement of a sum of subspaces is the intersection of the orthogonal complements:{{harvnb|Halmos|1957|loc=§12}}

$$\backslash biggl(\backslash sum\_i\; V\_i\backslash biggr)^\backslash perp\; =\; \backslash bigcap\_i\; V\_i^\backslash perp\; \backslash ,.$$

If the {{math|V_i}} are in addition closed, then

$$\backslash overline\{\backslash sum\_i\; V\_i^\backslash perp\; \backslash vphantom\backslash Big|\}\; =\; \backslash biggl(\backslash bigcap\_i\; V\_i\backslash biggr)^\backslash perp\; \backslash ,.$$

There is a well-developed spectral theory for self-adjoint operators in a Hilbert space, that is roughly analogous to the study of symmetric matrices over the reals or self-adjoint matrices over the complex numbers.A general account of spectral theory in Hilbert spaces can be found in {{harvtxt|Riesz|Sz.-Nagy|1990}}. A more sophisticated account in the language of C*-algebras is in {{harvtxt|Rudin|1973}} or {{harvtxt|Kadison|Ringrose|1997}} In the same sense, one can obtain a "diagonalization" of a self-adjoint operator as a suitable sum (actually an integral) of orthogonal projection operators.

The spectrum of an operator {{math|T}}, denoted {{math|σ(T)}}, is the set of complex numbers {{math|λ}} such that {{math|T − λ}} lacks a continuous inverse. If {{math|T}} is bounded, then the spectrum is always a compact set in the complex plane, and lies inside the disc {{math|{{abs|z}} ≤ {{norm|T}}}}. If {{math|T}} is self-adjoint, then the spectrum is real. In fact, it is contained in the interval {{math|[m, M]}} where

$$m\; =\; \backslash inf\_\{\backslash |x\backslash |=1\}\backslash langle\; Tx,\; x\backslash rangle\; \backslash ,,\backslash quad\; M\; =\; \backslash sup\_\{\backslash |x\backslash |=1\}\backslash langle\; Tx,\; x\backslash rangle\; \backslash ,.$$

Moreover, {{math|m}} and {{math|M}} are both actually contained within the spectrum.

The eigenspaces of an operator {{math|T}} are given by

$$H\_\backslash lambda\; =\; \backslash ker(T\; -\; \backslash lambda)\backslash ,.$$

Unlike with finite matrices, not every element of the spectrum of {{math|T}} must be an eigenvalue: the linear operator {{math|T − λ}} may only lack an inverse because it is not surjective. Elements of the spectrum of an operator in the general sense are known as spectral values. Since spectral values need not be eigenvalues, the spectral decomposition is often more subtle than in finite dimensions.

However, the spectral theorem of a self-adjoint operator {{math|T}} takes a particularly simple form if, in addition, {{math|T}} is assumed to be a compact operator. The spectral theorem for compact self-adjoint operators states:See, for instance, {{harvtxt|Riesz|Sz.-Nagy|1990|loc=Chapter VI}} or {{harvnb|Weidmann|1980|loc=Chapter 7}}. This result was already known to {{harvtxt|Schmidt|1908}} in the case of operators arising from integral kernels.

• A compact self-adjoint operator {{math|T}} has only countably (or finitely) many spectral values. The spectrum of {{math|T}} has no limit point in the complex plane except possibly zero. The eigenspaces of {{math|T}} decompose {{math|H}} into an orthogonal direct sum: $$H\; =\; \backslash bigoplus\_\{\backslash lambda\backslash in\backslash sigma(T)\}H\_\backslash lambda\; \backslash ,.$$ Moreover, if {{math|E_λ}} denotes the orthogonal projection onto the eigenspace {{math|H_λ}}, then $$T\; =\; \backslash sum\_\{\backslash lambda\backslash in\backslash sigma(T)\}\; \backslash lambda\; E\_\backslash lambda\; \backslash ,,$$ where the sum converges with respect to the norm on {{math|B(H)}}.

This theorem plays a fundamental role in the theory of integral equations, as many integral operators are compact, in particular those that arise from Hilbert–Schmidt operators.

The general spectral theorem for self-adjoint operators involves a kind of operator-valued Riemann–Stieltjes integral, rather than an infinite summation.{{harvnb|Riesz|Sz.-Nagy|1990|loc=§§107–108}} The spectral family associated to {{math|T}} associates to each real number λ an operator {{math|E_λ}}, which is the projection onto the nullspace of the operator {{math|(T − λ)⁺}}, where the positive part of a self-adjoint operator is defined by

$$A^+\; =\; \backslash tfrac\{1\}\{2\}\backslash Bigl(\backslash sqrt\{A^2\}\; +\; A\backslash Bigr)\; \backslash ,.$$

The operators {{math|E_λ}} are monotone increasing relative to the partial order defined on self-adjoint operators; the eigenvalues correspond precisely to the jump discontinuities. One has the spectral theorem, which asserts

$$T\; =\; \backslash int\_\backslash mathbb\{R\}\; \backslash lambda\backslash ,\; \backslash mathrm\{d\}E\_\backslash lambda\; \backslash ,.$$

The integral is understood as a Riemann–Stieltjes integral, convergent with respect to the norm on {{math|B(H)}}. In particular, one has the ordinary scalar-valued integral representation

$$\backslash langle\; Tx,\; y\backslash rangle\; =\; \backslash int\_\{\backslash R\}\; \backslash lambda\backslash ,\backslash mathrm\{d\}\backslash langle\; E\_\backslash lambda\; x,\; y\backslash rangle\; \backslash ,.$$

A somewhat similar spectral decomposition holds for normal operators, although because the spectrum may now contain non-real complex numbers, the operator-valued Stieltjes measure {{math|dE_λ}} must instead be replaced by a resolution of the identity.

A major application of spectral methods is the spectral mapping theorem, which allows one to apply to a self-adjoint operator {{math|T}} any continuous complex function {{math|f}} defined on the spectrum of {{math|T}} by forming the integral

$$f(T)\; =\; \backslash int\_\{\backslash sigma(T)\}\; f(\backslash lambda)\backslash ,\backslash mathrm\{d\}E\_\backslash lambda\; \backslash ,.$$

The resulting continuous functional calculus has applications in particular to pseudodifferential operators.{{harvnb|Shubin|1987}}

The spectral theory of unbounded self-adjoint operators is only marginally more difficult than for bounded operators. The spectrum of an unbounded operator is defined in precisely the same way as for bounded operators: {{math|λ}} is a spectral value if the resolvent operator

$$R\_\backslash lambda\; =\; (T\; -\; \backslash lambda)^\{-1\}$$

fails to be a well-defined continuous operator. The self-adjointness of {{math|T}} still guarantees that the spectrum is real. Thus the essential idea of working with unbounded operators is to look instead at the resolvent {{math|R_λ}} where {{math|λ}} is nonreal. This is a bounded normal operator, which admits a spectral representation that can then be transferred to a spectral representation of {{math|T}} itself. A similar strategy is used, for instance, to study the spectrum of the Laplace operator: rather than address the operator directly, one instead looks as an associated resolvent such as a Riesz potential or Bessel potential.

A precise version of the spectral theorem in this case is:{{harvnb|Rudin|1973|loc=Theorem 13.30}}

{{math theorem|math_statement=Given a densely defined self-adjoint operator {{math|T}} on a Hilbert space {{math|H}}, there corresponds a unique resolution of the identity {{math|E}} on the Borel sets of {{math|R}}, such that $$\backslash langle\; Tx,\; y\; \backslash rangle\; =\; \backslash int\_\backslash R\; \backslash lambda\; \backslash ,\; \backslash mathrm\{d\}E\_\{x,y\}(\backslash lambda)$$ for all {{math|x ∈ D(T)}} and {{math|y ∈ H}}. The spectral measure {{math|E}} is concentrated on the spectrum of {{math|T}}.}}

There is also a version of the spectral theorem that applies to unbounded normal operators.

In Gravity's Rainbow (1973), a novel by Thomas Pynchon, one of the characters is called "Sammy Hilbert-Spaess", a pun on "Hilbert Space". The novel refers also to Gödel's incompleteness theorems.{{cite book|title=Gravity's Rainbow|last=Pynchon|first=Thomas|publisher=Viking Press|year=1973|isbn=978-0143039945|pages=217, 275}}

{{portal|Mathematics}}

{{cols|colwidth=21em}}

• {{annotated link|Banach space}}
• {{annotated link|Fock space}}
• {{annotated link|Fundamental theorem of Hilbert spaces}}
• {{annotated link|Hadamard space}}
• {{annotated link|Hausdorff space}}
• {{annotated link|Hilbert algebra (disambiguation)|Hilbert algebra}}
• {{annotated link|Hilbert C*-module}}
• {{annotated link|Hilbert manifold}}
• {{annotated link|L-semi-inner product}}
• {{annotated link|Locally convex topological vector space}}
• {{annotated link|Operator theory}}
• {{annotated link|Operator topologies}}
• {{annotated link|Quantum state space}}
• {{annotated link|Rigged Hilbert space}}
• {{annotated link|Topological vector space}}

{{colend}}

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{{refend}}

{{Wikibooks|Functional Analysis/Hilbert spaces}}

{{Commons category}}

• {{springer|title=Hilbert space|id=p/h047380}}
• [http://mathworld.wolfram.com/HilbertSpace.html Hilbert space at Mathworld]
• [http://terrytao.wordpress.com/2009/01/17/254a-notes-5-hilbert-spaces/ 245B, notes 5: Hilbert spaces] by Terence Tao

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