Glossary of functional analysis

This is a glossary for the terminology in a mathematical field of functional analysis.

Throughout the article, unless stated otherwise, the base field of a vector space is the field of real numbers or that of complex numbers. Algebras are not assumed to be unital.

*

{{defn|1=*-homomorphism between involutive Banach algebras is an algebra homomorphism preserving *.}}

A

{{defn|1=Synonymous with "commutative"; e.g., an abelian Banach algebra means a commutative Banach algebra.}}

{{defn|1=The Anderson–Kadec theorem says a separable infinite-dimensional Fréchet space is isomorphic to $\mathbb{R}^{\mathbb{N}}$ .}}

{{defn|1=Alaoglu's theorem states that the closed unit ball in a normed space is compact in the weak-* topology.}}

{{defn|1=The adjoint of a bounded linear operator $T: H_1 \to H_2$ between Hilbert spaces is the bounded linear operator $T^* : H_2 \to H_1$ such that $\langle Tx, y \rangle = \langle x, T^* y \rangle$ for each $x \in H_1, y \in H_2$ .}}

{{defn|1=In a not-necessarily-unital Banach algebra, an approximate identity is a sequence or a net $\{ u_i \}$ of elements such that $u_i x \to x, x u_i \to x$ as $i \to \infty$ for each x in the algebra.}}

{{defn|1=A Banach space is said to have the approximation property if every compact operator is a limit of finite-rank operators.}}

B

{{defn|1=The Baire category theorem states that a complete metric space is a Baire space; if $U_i$ is a sequence of open dense subsets, then $\cap_1^{\infty} U_i$ is dense.}}

{{defn|no=1|1=A Banach space is a normed vector space that is complete as a metric space.}}

{{defn|no=2|1=A Banach algebra is a Banach space that has a structure of a possibly non-unital associative algebra such that

: $\|x y \| \le \|x\| \|y\|$ for every $x, y$ in the algebra.}}

{{defn|no=3|1=A Banach disc is a continuous linear image of a unit ball in a Banach space.}}

{{defn|1=A subset S of a vector space over real or complex numbers is balanced if $\lambda S \subset S$ for every scalar $\lambda$ of length at most one.}}

{{defn|no=1|A barrel in a topological vector space is a subset that is closed, convex, balanced and absorbing.}}

{{defn|no=2|A topological vector space is barrelled if every barrel is a neighborhood of zero (that is, contains an open neighborhood of zero).}}

{{defn|Bessel's inequality states: given an orthonormal set S and a vector x in a Hilbert space,

: $\sum_{u \in S} |\langle x, u \rangle|^2 \le \|x\|^2$ ,

where the equality holds if and only if S is an orthonormal basis; i.e., maximal orthonormal set.}}

{{defn|1=A bounded operator is a linear operator between Banach spaces for which the image of the unit ball is bounded.}}

{{defn|1=Two vectors x and y in a normed linear space are said to be Birkhoff orthogonal if $\| x + \lambda y \| \ge \|x\|$ for all scalars λ. If the normed linear space is a Hilbert space, then it is equivalent to the usual orthogonality.}}

C

{{defn|1=The Calkin algebra on a Hilbert space is the quotient of the algebra of all bounded operators on the Hilbert space by the ideal generated by compact operators.}}

{{defn|1=The Cauchy–Schwarz inequality states: for each pair of vectors $x, y$ in an inner-product space,

: $|\langle x, y \rangle| \le \|x\| \|y\|$ .}}

{{defn|no=1|1=The closed graph theorem states that a linear operator between Banach spaces is continuous (bounded) if and only if it has closed graph.}}

{{defn|no=2|1=A closed operator is a linear operator whose graph is closed.}}

{{defn|no=3|1=The closed range theorem says that a densely defined closed operator has closed image (range) if and only if the transpose of it has closed image.}}

{{defn|no=1|1=Another name for "centralizer"; i.e., the commutant of a subset S of an algebra is the algebra of the elements commuting with each element of S and is denoted by $S'$ .}}

{{defn|no=2|1=The von Neumann double commutant theorem states that a nondegenerate *-algebra $\mathfrak{M}$ of operators on a Hilbert space is a von Neumann algebra if and only if $\mathfrak{M}'' = \mathfrak{M}$ .}}

{{defn|1=A compact operator is a linear operator between Banach spaces for which the image of the unit ball is precompact.}}

{{defn|1=A C* algebra is an involutive Banach algebra satisfying $\|x^* x\| = \|x^*\| \|x\|$ .}}

{{defn|1=A locally convex space is a topological vector space whose topology is generated by convex subsets.}}

{{defn|1=Given a representation $(\pi, V)$ of a Banach algebra $A$ , a cyclic vector is a vector $v \in V$ such that $\pi(A)v$ is dense in $V$ .}}

D

{{defn|1=Philosophically, a direct integral is a continuous analog of a direct sum.}}

{{defn|no=1|The continuous dual of a topological vector space is the vector space of all the continuous linear functionals on the space.}}

{{defn|no=2|The algebraic dual of a topological vector space is the dual vector space of the underlying vector space.}}

E

F

{{defn|1=A factor is a von Neumann algebra with trivial center.}}

{{defn|1=A linear functional $\omega$ on an involutive algebra is faithful if $\omega(x^*x) \ne 0$ for each nonzero element $x$ in the algebra.}}

{{defn|1=A Fréchet space is a topological vector space whose topology is given by a countable family of seminorms (which makes it a metric space) and that is complete as a metric space.}}

{{defn|1=A Fredholm operator is a bounded operator such that it has closed range and the kernels of the operator and the adjoint have finite-dimension.}}

G

{{defn|no=1|1=The Gelfand–Mazur theorem states that a Banach algebra that is a division ring is the field of complex numbers.}}

{{defn|no=2|1=The Gelfand representation of a commutative Banach algebra $A$ with spectrum $\Omega(A)$ is the algebra homomorphism $F: A \to C_0(\Omega(A))$ , where $C_0(X)$ denotes the algebra of continuous functions on $X$ vanishing at infinity, that is given by $F(x)(\omega) = \omega(x)$ . It is a *-preserving isometric isomorphism if $A$ is a commutative C*-algebra.}}

H

{{defn|1=The Hahn–Banach theorem states: given a linear functional $\ell$ on a subspace of a complex vector space V, if the absolute value of $\ell$ is bounded above by a seminorm on V, then it extends to a linear functional on V still bounded by the seminorm. Geometrically, it is a generalization of the hyperplane separation theorem.}}

{{defn|1=A topological vector space is said to have the Heine–Borel property if every closed and bounded subset is compact. Riesz's lemma says a Banach space with the Heine–Borel property must be finite-dimensional.}}

{{defn|no=1|1=A Hilbert space is an inner product space that is complete as a metric space.}}

{{defn|no=2|1=In the Tomita–Takesaki theory, a (left or right) Hilbert algebra is a certain algebra with an involution.}}

{{defn|no=1|1=The Hilbert–Schmidt norm of a bounded operator $T$ on a Hilbert space is $\sum_i \|T e_i \|^2$ where $\{ e_i \}$ is an orthonormal basis of the Hilbert space.}}

{{defn|no=2|1=A Hilbert–Schmidt operator is a bounded operator with finite Hilbert–Schmidt norm.}}

I

{{defn|no=1|1=The index of a Fredholm operator $T : H_1 \to H_2$ is the integer $\operatorname{dim}(\operatorname{ker}(T^*)) - \operatorname{dim}(\operatorname{ker}(T))$ .}}

{{defn|The index group of a unital Banach algebra is the quotient group $G(A)/G_0(A)$ where $G(A)$ is the unit group of A and $G_0(A)$ the identity component of the group.}}

{{defn|no=1|1=An inner product on a real or complex vector space $V$ is a function $\langle \cdot, \cdot \rangle : V \times V \to \mathbb{R}$ such that for each $v, w \in V$ , (1) $x \mapsto \langle x, v \rangle$ is linear and (2) $\langle v, w \rangle = \overline{\langle w, v\rangle}$ where the bar means complex conjugate.}}

{{defn|no=2|1=An inner product space is a vector space equipped with an inner product.}}

{{defn|no=1|1=An involution of a Banach algebra A is an isometric endomorphism $A \to A, \, x \mapsto x^*$ that is conjugate-linear and such that $(xy)^* = (yx)^*$ .}}

{{defn|no=2|1=An involutive Banach algebra is a Banach algebra equipped with an involution.}}

{{defn|1=A linear isometry between normed vector spaces is a linear map preserving norm.}}

K

{{defn|1=A Köthe sequence space. For now, see https://mathoverflow.net/questions/361048/on-k%C3%B6the-sequence-spaces}}

{{defn|1=The Krein–Milman theorem states: a nonempty compact convex subset of a locally convex space has an extremal point.}}

L

{{defn|1=Linear Operators is a three-value book by Dunford and Schwartz.}}

{{defn|1=A locally convex algebra is an algebra whose underlying vector space is a locally convex space and whose multiplication is continuous with respect to the locally convex space topology.}}

M

N

{{defn|1=A representation $(\pi, V)$ of an algebra $A$ is said to be nondegenerate if for each vector $v \in V$ , there is an element $a \in A$ such that $\pi(a) v \ne 0$ .}}

{{defn|no=1|1=A norm on a vector space X is a real-valued function $\| \cdot \| : X \to \mathbb{R}$ such that for each scalar $a$ and vectors $x, y$ in $X$ , (1) $\| ax\| = |a| \| x \|$ , (2) (triangular inequality) $\| x + y \| \le \| x \| + \| y \|$ and (3) $\| x \| \ge 0$ where the equality holds only for $x = 0$ .}}

{{defn|no=2|1=A normed vector space is a real or complex vector space equipped with a norm $\| \cdot \|$ . It is a metric space with the distance function $d(x, y) = \| x - y \|$ .}}

O

{{defn|1=A one parameter group of a unital Banach algebra A is a continuous group homomorphism from $(\mathbb{R}, +)$ to the unit group of A.}}

{{defn|1=The open mapping theorem says a surjective continuous linear operator between Banach spaces is an open mapping.}}

{{defn|no=1|1=A subset S of a Hilbert space is orthonormal if, for each u, v in the set, $\langle u, v \rangle$ = 0 when $u \ne v$ and $= 1$ when $u = v$ .}}

{{defn|no=2|1=An orthonormal basis is a maximal orthonormal set (note: it is *not* necessarily a vector space basis.)}}

{{defn|no=1|1=Given a Hilbert space H and a closed subspace M, the orthogonal complement of M is the closed subspace $M^{\bot} = \{ x \in H | \langle x, y \rangle = 0, y \in M \}$ .}}

{{defn|no=2|1=In the notations above, the orthogonal projection $P$ onto M is a (unique) bounded operator on H such that $P^2 = P, P^* = P, \operatorname{im}(P) = M, \operatorname{ker}(P) = M^{\bot}.$ }}

P

{{defn|1=Parseval's identity states: given an orthonormal basis S in a Hilbert space, $\| x \|^2 = \sum_{u \in S} |\langle x, u \rangle|^2$ .Here, the part of the assertion is $\sum_{u \in S} \cdots$ is well-defined; i.e., when S is infinite, for countable totally ordered subsets $S' \subset S$ , $\sum_{u \in S'} \cdots$ is independent of $S'$ and $\sum_{u \in S} \cdots$ denotes the common value.}}

{{defn|1=A linear functional $\omega$ on an involutive Banach algebra is said to be positive if $\omega(x^* x) \ge 0$ for each element $x$ in the algebra.}}

{{defn|1=An operator T is called a projection if it is an idempotent; i.e., $T^2 = T$ .}}

Q

R

{{defn|1=A reflexive space is a topological vector space such that the natural map from the vector space to the second (topological) dual is an isomorphism.}}

{{defn|1=The resolvent of an element x of a unital Banach algebra is the complement in $\mathbb{C}$ of the spectrum of x.}}

S

{{defn|1=A self-adjoint operator is a bounded operator whose adjoint is itself. More generally, a closed densely defined operator is called self-adjoint if it coincides with the adjoint including the domain.}}

{{defn|1=A locally convex space is called semi-reflexive space if the canonical map to the second continuous dual is surjective.}}

{{defn|1=A separable Hilbert space is a Hilbert space admitting a finite or countable orthonormal basis.}}

{{defn|no=1|1=The spectrum of an element x of a unital Banach algebra is the set of complex numbers $\lambda$ such that $x - \lambda$ is not invertible.}}

{{defn|no=2|1=The spectrum of a commutative Banach algebra is the set of all characters (a homomorphism to $\mathbb{C}$ ) on the algebra.}}

{{defn|no=1|1=The spectral radius of an element x of a unital Banach algebra is $\sup_{\lambda} |\lambda|$ where the sup is over the spectrum of x.}}

{{defn|no=2|1=The spectral mapping theorem states: if x is an element of a unital Banach algebra and f is a holomorphic function in a neighborhood of the spectrum $\sigma(x)$ of x, then $f(\sigma(x)) = \sigma(f(x))$ , where $f(x)$ is an element of the Banach algebra defined via the Cauchy's integral formula.}}

{{defn|A linear operator T on a pre-Hilbert space is symmetric if $(Tx, y) = (x, Ty).$ }}

T

{{defn|no=1|1=See topological tensor product. Note it is still somewhat of an open problem to define or work out a correct tensor product of topological vector spaces, including Banach spaces.}}

{{defn|no=1|1=A topological vector space is a vector space equipped with a topology such that (1) the topology is Hausdorff and (2) the addition $(x, y) \mapsto x + y$ as well as scalar multiplication $(\lambda, x) \mapsto \lambda x$ are continuous.}}

{{defn|no=2|1=A linear map $f: E \to F$ is called a topological homomorphism if $f : E \to \operatorname{im}(f)$ is an open mapping.}}

{{defn|no=3|1=A sequence $\cdots \to E_{n -1} \to E_n \to E_{n+1} \to \cdots$ is called topologically exact if it is an exact sequence on the underlying vector spaces and, moreover, each $E_n \to E_{n+1}$ is a topological homomorphism.}}

U

{{defn|1=An unbounded operator is a partially defined linear operator, usually defined on a dense subspace.}}

{{defn|1=The uniform boundedness principle states: given a set of operators between Banach spaces, if $\sup_T |Tx| < \infty$ , sup over the set, for each x in the Banach space, then $\sup_T \|T\| < \infty$ .}}

{{defn|no=1|1=A unitary operator between Hilbert spaces is an invertible bounded linear operator such that the inverse is the adjoint of the operator.}}

{{defn|no=2|1=Two representations $(\pi_1, H_1), (\pi_2, H_2)$ of an involutive Banach algebra A on Hilbert spaces $H_1, H_2$ are said to be unitarily equivalent if there is a unitary operator $U: H_1 \to H_2$ such that $\pi_2(x) U = U \pi_1(x)$ for each x in A.}}

V

W

{{defn|1=A W*-algebra is a C*-algebra that admits a faithful representation on a Hilbert space such that the image of the representation is a von Neumann algebra.}}

References

Bourbaki, Espaces vectoriels topologiques
{{Citation | last1=Connes | first1=Alain | author1-link=Alain Connes | title=Non-commutative geometry | url=https://archive.org/details/noncommutativege0000conn | publisher=Academic Press | location=Boston, MA | isbn=978-0-12-185860-5 | year=1994 | url-access=registration }}
{{Conway A Course in Functional Analysis|edition=2}}
{{Dunford Schwartz Linear Operators Part 1 General Theory}}
{{Rudin Walter Functional Analysis|edition=2}}
M. Takesaki, Theory of Operator Algebras I, Springer, 2001, 2nd printing of the first edition 1979.
{{citation | last1=Yoshida| first1=Kôsaku | title=Functional Analysis | year=1980| publisher=Springer |edition=sixth}}

Glossary of functional analysis

*

A

B

C

D

E

F

G

H

I

K

L

M

N

O

P

Q

R

S

T

U

V

W

References

Further reading