Riesz representation theorem#The representation theorem for linear functionals on Cc.28X.29

Let $H$ be a Hilbert space over a field $\backslash mathbb\{F\},$ where $\backslash mathbb\{F\}$ is either the real numbers $\backslash R$ or the complex numbers $\backslash Complex.$ If $\backslash mathbb\{F\}\; =\; \backslash Complex$ (resp. if $\backslash mathbb\{F\}\; =\; \backslash R$) then $H$ is called a {{em|complex Hilbert space}} (resp. a {{em|real Hilbert space}}). Every real Hilbert space can be extended to be a dense subset of a unique (up to bijective isometry) complex Hilbert space, called its complexification, which is why Hilbert spaces are often automatically assumed to be complex. Real and complex Hilbert spaces have in common many, but by no means all, properties and results/theorems.

This article is intended for both mathematicians and physicists and will describe the theorem for both.

In both mathematics and physics, if a Hilbert space is assumed to be real (that is, if $\backslash mathbb\{F\}\; =\; \backslash R$) then this will usually be made clear. Often in mathematics, and especially in physics, unless indicated otherwise, "Hilbert space" is usually automatically assumed to mean "complex Hilbert space." Depending on the author, in mathematics, "Hilbert space" usually means either (1) a complex Hilbert space, or (2) a real {{em|or}} complex Hilbert space.

By definition, an Antilinear map (also called a {{em|conjugate-linear map}}) $f\; :\; H\; \backslash to\; Y$ is a map between vector spaces that is {{em|additive}}:

$$f(x\; +\; y)\; =\; f(x)\; +\; f(y)\; \backslash quad\; \backslash text\{\; for\; all\; \}\; x,\; y\; \backslash in\; H,$$

and {{em|antilinear}} (also called {{em|conjugate-linear}} or {{em|conjugate-homogeneous}}):

$$f(c\; x)\; =\; \backslash overline\{c\}\; f(x)\; \backslash quad\; \backslash text\{\; for\; all\; \}\; x\; \backslash in\; H\; \backslash text\{\; and\; all\; scalar\; \}\; c\; \backslash in\; \backslash mathbb\{F\},$$

where $\backslash overline\{c\}$ is the conjugate of the complex number $c\; =\; a\; +\; b\; i$, given by $\backslash overline\{c\}\; =\; a\; -\; b\; i$.

In contrast, a map $f\; :\; H\; \backslash to\; Y$ is linear if it is additive and Homogeneous function:

$$f(c\; x)\; =\; c\; f(x)\; \backslash quad\; \backslash text\{\; for\; all\; \}\; x\; \backslash in\; H\; \backslash quad\; \backslash text\{\; and\; all\; scalars\; \}\; c\; \backslash in\; \backslash mathbb\{F\}.$$

Every constant $0$ map is always both linear and antilinear. If $\backslash mathbb\{F\}\; =\; \backslash R$ then the definitions of linear maps and antilinear maps are completely identical. A linear map from a Hilbert space into a Banach space (or more generally, from any Banach space into any topological vector space) is continuous if and only if it is bounded; the same is true of antilinear maps. The inverse of any antilinear (resp. linear) bijection is again an antilinear (resp. linear) bijection. The composition of two {{em|anti}}linear maps is a {{em|linear}} map.

Continuous dual and anti-dual spaces

A {{em|functional}} on $H$ is a function $H\; \backslash to\; \backslash mathbb\{F\}$ whose codomain is the underlying scalar field $\backslash mathbb\{F\}.$

Denote by $H^*$ (resp. by $\backslash overline\{H\}^*)$ the set of all continuous linear (resp. continuous antilinear) functionals on $H,$ which is called the {{em|(continuous) dual space}} (resp. the {{em|(continuous) anti-dual space}}) of $H.${{sfn|Trèves|2006|pp=112–123}}

If $\backslash mathbb\{F\}\; =\; \backslash R$ then linear functionals on $H$ are the same as antilinear functionals and consequently, the same is true for such continuous maps: that is, $H^*\; =\; \backslash overline\{H\}^*.$

One-to-one correspondence between linear and antilinear functionals

Given any functional $f\; ~:~\; H\; \backslash to\; \backslash mathbb\{F\},$ the {{em|conjugate of $f$}} is the functional

$$\backslash begin\{alignat\}\{4\}$$

\overline{f} : \,& H && \to \,&& \mathbb{F} \\

& h && \mapsto\,&& \overline{f(h)}. \\

\end{alignat}

This assignment is most useful when $\backslash mathbb\{F\}\; =\; \backslash Complex$ because if $\backslash mathbb\{F\}\; =\; \backslash R$ then $f\; =\; \backslash overline\{f\}$ and the assignment $f\; \backslash mapsto\; \backslash overline\{f\}$ reduces down to the identity map.

The assignment $f\; \backslash mapsto\; \backslash overline\{f\}$ defines an antilinear bijective correspondence from the set of

:all functionals (resp. all linear functionals, all continuous linear functionals $H^*$) on $H,$

onto the set of

:all functionals (resp. all {{em|anti}}linear functionals, all continuous {{em|anti}}linear functionals $\backslash overline\{H\}^*$) on $H.$

The Hilbert space $H$ has an associated inner product $H\; \backslash times\; H\; \backslash to\; \backslash mathbb\{F\}$ valued in $H$'s underlying scalar field $\backslash mathbb\{F\}$ that is linear in one coordinate and antilinear in the other (as specified below).

If $H$ is a complex Hilbert space ($\backslash mathbb\{F\}\; =\; \backslash Complex$), then there is a crucial difference between the notations prevailing in mathematics versus physics, regarding which of the two variables is linear.

However, for real Hilbert spaces ($\backslash mathbb\{F\}\; =\; \backslash R$), the inner product is a symmetric map that is linear in each coordinate (bilinear), so there can be no such confusion.

In mathematics, the inner product on a Hilbert space $H$ is often denoted by $\backslash left\backslash langle\; \backslash cdot\backslash ,,\; \backslash cdot\; \backslash right\backslash rangle$ or $\backslash left\backslash langle\; \backslash cdot\backslash ,,\; \backslash cdot\; \backslash right\backslash rangle\_H$ while in physics, the bra–ket notation $\backslash left\backslash langle\; \backslash cdot\; \backslash mid\; \backslash cdot\; \backslash right\backslash rangle$ or $\backslash left\backslash langle\; \backslash cdot\; \backslash mid\; \backslash cdot\; \backslash right\backslash rangle\_H$ is typically used. In this article, these two notations will be related by the equality:

$$\backslash left\backslash langle\; x,\; y\; \backslash right\backslash rangle\; :=\; \backslash left\backslash langle\; y\; \backslash mid\; x\; \backslash right\backslash rangle\; \backslash quad\; \backslash text\{\; for\; all\; \}\; x,\; y\; \backslash in\; H.$$These have the following properties:

1. The map $\backslash left\backslash langle\; \backslash cdot\backslash ,,\; \backslash cdot\; \backslash right\backslash rangle$ is linear in its first coordinate; equivalently, the map $\backslash left\backslash langle\; \backslash cdot\; \backslash mid\; \backslash cdot\; \backslash right\backslash rangle$ is linear in its second coordinate. That is, for fixed $y\; \backslash in\; H,$ the map

   $\backslash left\backslash langle\; \backslash ,y\backslash mid\; \backslash cdot\backslash ,\; \backslash right\backslash rangle\; =\; \backslash left\backslash langle\; \backslash ,\backslash cdot\backslash ,,\; y\backslash ,\; \backslash right\backslash rangle\; :\; H\; \backslash to\; \backslash mathbb\{F\}$ with $h\; \backslash mapsto\; \backslash left\backslash langle\; \backslash ,y\backslash mid\; h\backslash ,\; \backslash right\backslash rangle\; =\; \backslash left\backslash langle\; \backslash ,h,\; y\backslash ,\; \backslash right\backslash rangle$

   is a linear functional on $H.$ This linear functional is continuous, so $\backslash left\backslash langle\; \backslash ,y\backslash mid\backslash cdot\backslash ,\; \backslash right\backslash rangle\; =\; \backslash left\backslash langle\; \backslash ,\backslash cdot,\; y\backslash ,\; \backslash right\backslash rangle\; \backslash in\; H^*.$

2. The map $\backslash left\backslash langle\; \backslash cdot\backslash ,,\; \backslash cdot\; \backslash right\backslash rangle$ is antilinear in its {{em|second}} coordinate; equivalently, the map $\backslash left\backslash langle\; \backslash cdot\; \backslash mid\; \backslash cdot\; \backslash right\backslash rangle$ is antilinear in its {{em|first}} coordinate. That is, for fixed $y\; \backslash in\; H,$ the map

   $\backslash left\backslash langle\; \backslash ,\backslash cdot\backslash mid\; y\backslash ,\; \backslash right\backslash rangle\; =\; \backslash left\backslash langle\; \backslash ,y,\; \backslash cdot\backslash ,\; \backslash right\backslash rangle\; :\; H\; \backslash to\; \backslash mathbb\{F\}$ with $h\; \backslash mapsto\; \backslash left\backslash langle\; \backslash ,h\backslash mid\; y\backslash ,\; \backslash right\backslash rangle\; =\; \backslash left\backslash langle\; \backslash ,y,\; h\backslash ,\; \backslash right\backslash rangle$

   is an antilinear functional on $H.$ This antilinear functional is continuous, so $\backslash left\backslash langle\; \backslash ,\backslash cdot\backslash mid\; y\backslash ,\; \backslash right\backslash rangle\; =\; \backslash left\backslash langle\; \backslash ,y,\; \backslash cdot\backslash ,\; \backslash right\backslash rangle\; \backslash in\; \backslash overline\{H\}^*.$

In computations, one must consistently use either the mathematics notation $\backslash left\backslash langle\; \backslash cdot\backslash ,,\; \backslash cdot\; \backslash right\backslash rangle$, which is (linear, antilinear); or the physics notation $\backslash left\backslash langle\; \backslash cdot\; \backslash mid\; \backslash cdot\; \backslash right\backslash rangle$, which is (antilinear | linear).

If $x\; =\; y$ then $\backslash langle\; \backslash ,x\backslash mid\; x\backslash ,\; \backslash rangle\; =\; \backslash langle\; \backslash ,x,\; x\backslash ,\; \backslash rangle$ is a non-negative real number and the map

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

defines a canonical norm on $H$ that makes $H$ into a normed space.{{sfn|Trèves|2006|pp=112–123}}

As with all normed spaces, the (continuous) dual space $H^*$ carries a canonical norm, called the {{em|dual norm}}, that is defined by{{sfn|Trèves|2006|pp=112–123}}

$$\backslash |f\backslash |\_\{H^*\}\; ~:=~\; \backslash sup\_\{\backslash |x\backslash |\; \backslash leq\; 1,\; x\; \backslash in\; H\}\; |f(x)|\; \backslash quad\; \backslash text\{\; for\; every\; \}\; f\; \backslash in\; H^*.$$

The canonical norm on the (continuous) anti-dual space $\backslash overline\{H\}^*,$ denoted by $\backslash |f\backslash |\_\{\backslash overline\{H\}^*\},$ is defined by using this same equation:{{sfn|Trèves|2006|pp=112–123}}

$$\backslash |f\backslash |\_\{\backslash overline\{H\}^*\}\; ~:=~\; \backslash sup\_\{\backslash |x\backslash |\; \backslash leq\; 1,\; x\; \backslash in\; H\}\; |f(x)|\; \backslash quad\; \backslash text\{\; for\; every\; \}\; f\; \backslash in\; \backslash overline\{H\}^*.$$

This canonical norm on $H^*$ satisfies the parallelogram law, which means that the polarization identity can be used to define a {{em|canonical inner product on $H^*,$}} which this article will denote by the notations

$$\backslash left\backslash langle\; f,\; g\; \backslash right\backslash rangle\_\{H^*\}\; :=\; \backslash left\backslash langle\; g\; \backslash mid\; f\; \backslash right\backslash rangle\_\{H^*\},$$

where this inner product turns $H^*$ into a Hilbert space. There are now two ways of defining a norm on $H^*:$ the norm induced by this inner product (that is, the norm defined by $f\; \backslash mapsto\; \backslash sqrt\{\backslash left\backslash langle\; f,\; f\; \backslash right\backslash rangle\_\{H^*\}\}$) and the usual dual norm (defined as the supremum over the closed unit ball). These norms are the same; explicitly, this means that the following holds for every $f\; \backslash in\; H^*:$

$$\backslash sup\_\{\backslash |x\backslash |\; \backslash leq\; 1,\; x\; \backslash in\; H\}\; |f(x)|\; =\; \backslash |f\backslash |\_\{H^*\}\; ~=~\; \backslash sqrt\{\backslash langle\; f,\; f\; \backslash rangle\_\{H^*\}\}\; ~=~\; \backslash sqrt\{\backslash langle\; f\; \backslash mid\; f\; \backslash rangle\_\{H^*\}\}.$$

As will be described later, the Riesz representation theorem can be used to give an equivalent definition of the canonical norm and the canonical inner product on $H^*.$

The same equations that were used above can also be used to define a norm and inner product on $H$'s anti-dual space $\backslash overline\{H\}^*.${{sfn|Trèves|2006|pp=112–123}}

Canonical isometry between the dual and antidual

The complex conjugate $\backslash overline\{f\}$ of a functional $f,$ which was defined above, satisfies

$$\backslash |f\backslash |\_\{H^*\}\; ~=~\; \backslash left\backslash |\backslash overline\{f\}\backslash right\backslash |\_\{\backslash overline\{H\}^*\}\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash quad\; \backslash left\backslash |\backslash overline\{g\}\backslash right\backslash |\_\{H^*\}\; ~=~\; \backslash |g\backslash |\_\{\backslash overline\{H\}^*\}$$

for every $f\; \backslash in\; H^*$ and every $g\; \backslash in\; \backslash overline\{H\}^*.$

This says exactly that the canonical antilinear bijection defined by

$$\backslash begin\{alignat\}\{4\}$$

\operatorname{Cong} :\;&& H^* &&\;\to \;& \overline{H}^* \\[0.3ex]

&& f &&\;\mapsto\;& \overline{f} \\

\end{alignat}

as well as its inverse $\backslash operatorname\{Cong\}^\{-1\}\; ~:~\; \backslash overline\{H\}^*\; \backslash to\; H^*$ are antilinear isometries and consequently also homeomorphisms.

The inner products on the dual space $H^*$ and the anti-dual space $\backslash overline\{H\}^*,$ denoted respectively by $\backslash langle\; \backslash ,\backslash cdot\backslash ,,\; \backslash ,\backslash cdot\backslash ,\; \backslash rangle\_\{H^*\}$ and $\backslash langle\; \backslash ,\backslash cdot\backslash ,,\; \backslash ,\backslash cdot\backslash ,\; \backslash rangle\_\{\backslash overline\{H\}^*\},$ are related by

$$\backslash langle\; \backslash ,\backslash overline\{f\}\backslash ,\; |\; \backslash ,\backslash overline\{g\}\backslash ,\; \backslash rangle\_\{\backslash overline\{H\}^*\}\; =\; \backslash overline\{\backslash langle\; \backslash ,f\backslash ,\; |\; \backslash ,g\backslash ,\; \backslash rangle\_\{H^*\}\}\; =\; \backslash langle\; \backslash ,g\backslash ,\; |\; \backslash ,f\backslash ,\; \backslash rangle\_\{H^*\}\; \backslash qquad\; \backslash text\{\; for\; all\; \}\; f,\; g\; \backslash in\; H^*$$

and

$$\backslash langle\; \backslash ,\backslash overline\{f\}\backslash ,\; |\; \backslash ,\backslash overline\{g\}\backslash ,\; \backslash rangle\_\{H^*\}\; =\; \backslash overline\{\backslash langle\; \backslash ,f\backslash ,\; |\; \backslash ,g\backslash ,\; \backslash rangle\_\{\backslash overline\{H\}^*\}\}\; =\; \backslash langle\; \backslash ,g\backslash ,\; |\; \backslash ,f\backslash ,\; \backslash rangle\_\{\backslash overline\{H\}^*\}\; \backslash qquad\; \backslash text\{\; for\; all\; \}\; f,\; g\; \backslash in\; \backslash overline\{H\}^*.$$

If $\backslash mathbb\{F\}\; =\; \backslash R$ then $H^*\; =\; \backslash overline\{H\}^*$ and this canonical map $\backslash operatorname\{Cong\}\; :\; H^*\; \backslash to\; \backslash overline\{H\}^*$ reduces down to the identity map.

Two vectors $x$ and $y$ are {{em|orthogonal}} if $\backslash langle\; x,\; y\; \backslash rangle\; =\; 0,$ which happens if and only if $\backslash |y\backslash |\; \backslash leq\; \backslash |y\; +\; s\; x\backslash |$ for all scalars $s.${{sfn|Rudin|1991|pp=306-312}} The orthogonal complement of a subset $X\; \backslash subseteq\; H$ is

$$X^\{\backslash bot\}\; :=\; \backslash \{\; \backslash ,y\; \backslash in\; H\; :\; \backslash langle\; y,\; x\; \backslash rangle\; =\; 0\; \backslash text\{\; for\; all\; \}\; x\; \backslash in\; X\backslash ,\; \backslash \},$$

which is always a closed vector subspace of $H.$

The Hilbert projection theorem guarantees that for any nonempty closed convex subset $C$ of a Hilbert space there exists a unique vector $m\; \backslash in\; C$ such that $\backslash |m\backslash |\; =\; \backslash inf\_\{c\; \backslash in\; C\}\; \backslash |c\backslash |;$ that is, $m\; \backslash in\; C$ is the (unique) global minimum point of the function $C\; \backslash to\; [0,\; \backslash infty)$ defined by $c\; \backslash mapsto\; \backslash |c\backslash |.$

{{Math theorem

|name = {{visible anchor|Riesz representation theorem}}

|math_statement =

Let $H$ be a Hilbert space whose inner product $\backslash left\backslash langle\; x,\; y\; \backslash right\backslash rangle$ is linear in its {{em|first}} argument and antilinear in its second argument and let $\backslash langle\; y\; \backslash mid\; x\; \backslash rangle\; :=\; \backslash langle\; x,\; y\; \backslash rangle$ be the corresponding physics notation. For every continuous linear functional $\backslash varphi\; \backslash in\; H^*,$ there exists a unique vector $f\_\{\backslash varphi\}\; \backslash in\; H,$ called the {{em|{{visible anchor|Riesz representation|Riesz representative}} of $\backslash varphi,$}} such that{{harvnb|Roman|2008|loc=p. 351 Theorem 13.32}}

$$\backslash varphi(x)\; =\; \backslash left\backslash langle\; x,\; f\_\{\backslash varphi\}\; \backslash right\backslash rangle\; =\; \backslash left\backslash langle\; f\_\backslash varphi\; \backslash mid\; x\; \backslash right\backslash rangle\; \backslash quad\; \backslash text\{\; for\; all\; \}\; x\; \backslash in\; H.$$

Importantly for {{em|complex}} Hilbert spaces, $f\_\{\backslash varphi\}$ is always located in the {{em|antilinear}} coordinate of the inner product.

Furthermore, the length of the representation vector is equal to the norm of the functional:

$$\backslash left\backslash |f\_\backslash varphi\backslash right\backslash |\_H\; =\; \backslash |\backslash varphi\backslash |\_\{H^*\},$$ and $f\_\{\backslash varphi\}$ is the unique vector $f\_\{\backslash varphi\}\; \backslash in\; \backslash left(\backslash ker\; \backslash varphi\backslash right)^\{\backslash bot\}$ with $\backslash varphi\backslash left(f\_\{\backslash varphi\}\backslash right)\; =\; \backslash |\backslash varphi\backslash |^2.$

It is also the unique element of minimum norm in $C\; :=\; \backslash varphi^\{-1\}\backslash left(\backslash |\backslash varphi\backslash |^2\backslash right)$; that is to say, $f\_\{\backslash varphi\}$ is the unique element of $C$ satisfying $\backslash left\backslash |f\_\{\backslash varphi\}\backslash right\backslash |\; =\; \backslash inf\_\{c\; \backslash in\; C\}\; \backslash |c\backslash |.$

Moreover, any non-zero $q\; \backslash in\; (\backslash ker\; \backslash varphi)^\{\backslash bot\}$ can be written as $q\; =\; \backslash left(\backslash |q\backslash |^2\; /\backslash ,\; \backslash overline\{\backslash varphi(q)\}\backslash right)\backslash \; f\_\{\backslash varphi\}.$

}}

{{Math theorem

| name = Corollary

| math_statement = The {{em|canonical map from $H$ into its dual}} $H^*${{sfn|Trèves|2006|pp=112–123}} is the injective Antilinear map isometry{{sfn|Trèves|2006|pp=112–123}}

$$\backslash begin\{alignat\}\{4\}$$

\Phi :\;&& H &&\;\to \;& H^* \\[0.3ex]

&& y &&\;\mapsto\;& \langle \,\cdot\,, y \rangle = \langle y | \,\cdot\, \rangle \\

\end{alignat}

The Riesz representation theorem states that this map is surjective (and thus bijective) when $H$ is complete and that its inverse is the bijective isometric antilinear isomorphism

$$\backslash begin\{alignat\}\{4\}$$

\Phi^{-1} :\;&& H^* &&\;\to \;& H \\[0.3ex]

&& \varphi &&\;\mapsto\;& f_{\varphi} \\

\end{alignat}.

Consequently, {{em|every}} continuous linear functional on the Hilbert space $H$ can be written uniquely in the form $\backslash langle\; y\backslash ,\; |\; \backslash ,\backslash cdot\backslash ,\; \backslash rangle${{sfn|Trèves|2006|pp=112–123}} where $\backslash |\backslash langle\; y\backslash ,|\; \backslash cdot\; \backslash rangle\backslash |\_\{H^*\}\; =\; \backslash |y\backslash |\_H$ for every $y\; \backslash in\; H.$

The assignment $y\; \backslash mapsto\; \backslash langle\; y,\; \backslash cdot\; \backslash rangle\; =\; \backslash langle\; \backslash cdot\backslash ,|\; \backslash ,y\; \backslash rangle$ can also be viewed as a bijective {{em|linear}} isometry $H\; \backslash to\; \backslash overline\{H\}^*$ into the anti-dual space of $H,${{sfn|Trèves|2006|pp=112–123}} which is the complex conjugate vector space of the continuous dual space $H^*.$

The inner products on $H$ and $H^*$ are related by

$$\backslash left\backslash langle\; \backslash Phi\; h,\; \backslash Phi\; k\; \backslash right\backslash rangle\_\{H^*\}\; =\; \backslash overline\{\backslash langle\; h,\; k\; \backslash rangle\}\_H\; =\; \backslash langle\; k,\; h\; \backslash rangle\_H\; \backslash quad\; \backslash text\{\; for\; all\; \}\; h,\; k\; \backslash in\; H$$

and similarly,

$$\backslash left\backslash langle\; \backslash Phi^\{-1\}\; \backslash varphi,\; \backslash Phi^\{-1\}\; \backslash psi\; \backslash right\backslash rangle\_H\; =\; \backslash overline\{\backslash langle\; \backslash varphi,\; \backslash psi\; \backslash rangle\}\_\{H^*\}\; =\; \backslash left\backslash langle\; \backslash psi,\; \backslash varphi\; \backslash right\backslash rangle\_\{H^*\}\; \backslash quad\; \backslash text\{\; for\; all\; \}\; \backslash varphi,\; \backslash psi\; \backslash in\; H^*.$$

The set $C\; :=\; \backslash varphi^\{-1\}\backslash left(\backslash |\backslash varphi\backslash |^2\backslash right)$ satisfies $C\; =\; f\_\{\backslash varphi\}\; +\; \backslash ker\; \backslash varphi$ and $C\; -\; f\_\{\backslash varphi\}\; =\; \backslash ker\; \backslash varphi$ so when $f\_\{\backslash varphi\}\; \backslash neq\; 0$ then $C$ can be interpreted as being the affine hyperplane that is parallel to the vector subspace $\backslash ker\; \backslash varphi$ and contains $f\_\{\backslash varphi\}.$

For $y\; \backslash in\; H,$ the physics notation for the functional $\backslash Phi(y)\; \backslash in\; H^*$ is the bra $\backslash langle\; y\; |,$ where explicitly this means that $\backslash langle\; y\; |\; :=\; \backslash Phi(y),$ which complements the ket notation $|\; y\; \backslash rangle$ defined by $|\; y\; \backslash rangle\; :=\; y.$

In the mathematical treatment of quantum mechanics, the theorem can be seen as a justification for the popular bra–ket notation. The theorem says that, every bra $\backslash langle\backslash psi\backslash ,|$ has a corresponding ket $|\backslash ,\backslash psi\backslash rangle,$ and the latter is unique.

}}

Historically, the theorem is often attributed simultaneously to Riesz and Fréchet in 1907 (see references).

{{collapse top|title=Proof{{sfn|Rudin|1991|pp=307−309}}|left=true}}

Let $\backslash mathbb\{F\}$ denote the underlying scalar field of $H.$

{{em|Proof of norm formula:}}

Fix $y\; \backslash in\; H.$

Define $\backslash Lambda\; :\; H\; \backslash to\; \backslash mathbb\{F\}$ by $\backslash Lambda(z)\; :=\; \backslash langle\; \backslash ,y\backslash ,\; |\; \backslash ,z\backslash ,\; \backslash rangle,$ which is a linear functional on $H$ since $z$ is in the linear argument.

By the Cauchy–Schwarz inequality,

$$|\backslash Lambda(z)|\; =\; |\backslash langle\; \backslash ,y\backslash ,\; |\; \backslash ,z\backslash ,\; \backslash rangle|\; \backslash leq\; \backslash |y\backslash |\; \backslash |z\backslash |$$

which shows that $\backslash Lambda$ is bounded (equivalently, continuous) and that $\backslash |\backslash Lambda\backslash |\; \backslash leq\; \backslash |y\backslash |.$

It remains to show that $\backslash |y\backslash |\; \backslash leq\; \backslash |\backslash Lambda\backslash |.$

By using $y$ in place of $z,$ it follows that

$$\backslash |y\backslash |^2\; =\; \backslash langle\; \backslash ,y\backslash ,\; |\; \backslash ,y\backslash ,\; \backslash rangle\; =\; \backslash Lambda\; y\; =\; |\backslash Lambda(y)|\; \backslash leq\; \backslash |\backslash Lambda\backslash |\; \backslash |y\backslash |$$

(the equality $\backslash Lambda\; y\; =\; |\backslash Lambda(y)|$ holds because $\backslash Lambda\; y\; =\; \backslash |y\backslash |^2\; \backslash geq\; 0$ is real and non-negative).

Thus that $\backslash |\backslash Lambda\backslash |\; =\; \backslash |y\backslash |.$

$\backslash blacksquare$

The proof above did not use the fact that $H$ is complete, which shows that the formula for the norm $\backslash |\backslash langle\; \backslash ,y\backslash ,\; |\; \backslash ,\backslash cdot\backslash ,\; \backslash rangle\backslash |\_\{H^*\}\; =\; \backslash |y\backslash |\_H$ holds more generally for all inner product spaces.

{{hr|1}}

{{em|Proof that a Riesz representation of $\backslash varphi$ is unique:}}

Suppose $f,\; g\; \backslash in\; H$ are such that $\backslash varphi(z)\; =\; \backslash langle\; \backslash ,f\backslash ,\; |\; \backslash ,z\backslash ,\; \backslash rangle$ and $\backslash varphi(z)\; =\; \backslash langle\; \backslash ,g\backslash ,\; |\; \backslash ,z\backslash ,\; \backslash rangle$ for all $z\; \backslash in\; H.$

Then

$$\backslash langle\; \backslash ,f\; -\; g\backslash ,\; |\; \backslash ,z\backslash ,\; \backslash rangle\; =\; \backslash langle\; \backslash ,f\backslash ,\; |\; \backslash ,z\backslash ,\; \backslash rangle\; -\; \backslash langle\; \backslash ,g\backslash ,\; |\; \backslash ,z\backslash ,\; \backslash rangle\; =\; \backslash varphi(z)\; -\; \backslash varphi(z)\; =\; 0\; \backslash quad\; \backslash text\{\; for\; all\; \}\; z\; \backslash in\; H$$

which shows that $\backslash Lambda\; :=\; \backslash langle\; \backslash ,f\; -\; g\backslash ,\; |\; \backslash ,\backslash cdot\backslash ,\; \backslash rangle$ is the constant $0$ linear functional.

Consequently $0\; =\; \backslash |\backslash langle\; \backslash ,f\; -\; g\backslash ,\; |\; \backslash ,\backslash cdot\backslash ,\; \backslash rangle\backslash |\; =\; \backslash |f\; -\; g\backslash |,$ which implies that $f\; -\; g\; =\; 0.$

$\backslash blacksquare$

{{hr|1}}

{{em|Proof that a vector $f\_\{\backslash varphi\}$ representing $\backslash varphi$ exists:}}

Let $K\; :=\; \backslash ker\; \backslash varphi\; :=\; \backslash \{\; m\; \backslash in\; H\; :\; \backslash varphi(m)\; =\; 0\; \backslash \}.$

If $K\; =\; H$ (or equivalently, if $\backslash varphi\; =\; 0$) then taking $f\_\{\backslash varphi\}\; :=\; 0$ completes the proof so assume that $K\; \backslash neq\; H$ and $\backslash varphi\; \backslash neq\; 0.$

The continuity of $\backslash varphi$ implies that $K$ is a closed subspace of $H$ (because $K\; =\; \backslash varphi^\{-1\}(\backslash \{\; 0\; \backslash \})$ and $\backslash \{\; 0\; \backslash \}$ is a closed subset of $\backslash mathbb\{F\}$).

Let

$$K^\{\backslash bot\}\; :=\; \backslash \{\; v\; \backslash in\; H\; ~:~\; \backslash langle\; \backslash ,v\backslash ,\; |\; \backslash ,k\backslash ,\; \backslash rangle\; =\; 0\; ~\; \backslash text\{\; for\; all\; \}\; k\; \backslash in\; K\backslash \}$$

denote the orthogonal complement of $K$ in $H.$

Because $K$ is closed and $H$ is a Hilbert space,Showing that there is a non-zero vector $v$ in $K^\{\backslash bot\}$ relies on the continuity of $\backslash phi$ and the Cauchy completeness of $H.$ This is the only place in the proof in which these properties are used. $H$ can be written as the direct sum $H\; =\; K\; \backslash oplus\; K^\{\backslash bot\}$Technically, $H\; =\; K\; \backslash oplus\; K^\{\backslash bot\}$ means that the addition map $K\; \backslash times\; K^\{\backslash bot\}\; \backslash to\; H$ defined by $(k,\; p)\; \backslash mapsto\; k\; +\; p$ is a surjective linear isomorphism and homeomorphism. See the article on complemented subspaces for more details. (a proof of this is given in the article on the Hilbert projection theorem).

Because $K\; \backslash neq\; H,$ there exists some non-zero $p\; \backslash in\; K^\{\backslash bot\}.$

For any $h\; \backslash in\; H,$

$$\backslash varphi[(\backslash varphi\; h)\; p\; -\; (\backslash varphi\; p)\; h]$$

~=~ \varphi[(\varphi h) p] - \varphi[(\varphi p) h]

~=~ (\varphi h) \varphi p - (\varphi p) \varphi h

= 0,

which shows that $(\backslash varphi\; h)\; p\; -\; (\backslash varphi\; p)\; h\; ~\backslash in~\; \backslash ker\; \backslash varphi\; =\; K,$ where now $p\; \backslash in\; K^\{\backslash bot\}$ implies

$$0\; =\; \backslash langle\; \backslash ,p\backslash ,\; |\; \backslash ,(\backslash varphi\; h)\; p\; -\; (\backslash varphi\; p)\; h\backslash ,\; \backslash rangle$$

~=~ \langle \,p\, | \,(\varphi h) p \, \rangle - \langle \,p\, | \,(\varphi p) h\, \rangle

~=~ (\varphi h) \langle \,p\, | \,p \, \rangle - (\varphi p) \langle \,p\, | \,h\, \rangle.

Solving for $\backslash varphi\; h$ shows that

$$\backslash varphi\; h$$

= \frac{(\varphi p) \langle \,p\, | \,h\, \rangle}{\|p\|^2}

= \left\langle \,\frac{\overline{\varphi p}}{\|p\|^2} p\, \Bigg| \,h\, \right\rangle \quad \text{ for every } h \in H,

which proves that the vector $f\_\{\backslash varphi\}\; :=\; \backslash frac\{\backslash overline\{\backslash varphi\; p\}\}\{\backslash |p\backslash |^2\}\; p$ satisfies

$\backslash varphi\; h\; =\; \backslash langle\; \backslash ,f\_\{\backslash varphi\}\backslash ,\; |\; \backslash ,h\backslash ,\; \backslash rangle\; \backslash text\{\; for\; every\; \}\; h\; \backslash in\; H.$

Applying the norm formula that was proved above with $y\; :=\; f\_\{\backslash varphi\}$ shows that $\backslash |\backslash varphi\backslash |\_\{H^*\}\; =\; \backslash left\backslash |\backslash left\backslash langle\; \backslash ,f\_\{\backslash varphi\}\backslash ,\; |\; \backslash ,\backslash cdot\backslash ,\; \backslash right\backslash rangle\backslash right\backslash |\_\{H^*\}\; =\; \backslash left\backslash |f\_\{\backslash varphi\}\backslash right\backslash |\_H.$

Also, the vector $u\; :=\; \backslash frac\{p\}\{\backslash |p\backslash |\}$ has norm $\backslash |u\backslash |\; =\; 1$ and satisfies $f\_\{\backslash varphi\}\; :=\; \backslash overline\{\backslash varphi(u)\}\; u.$

$\backslash blacksquare$

{{hr|1}}

It can now be deduced that $K^\{\backslash bot\}$ is $1$-dimensional when $\backslash varphi\; \backslash neq\; 0.$

Let $q\; \backslash in\; K^\{\backslash bot\}$ be any non-zero vector. Replacing $p$ with $q$ in the proof above shows that the vector $g\; :=\; \backslash frac\{\backslash overline\{\backslash varphi\; q\}\}\{\backslash |q\backslash |^2\}\; q$ satisfies $\backslash varphi(h)\; =\; \backslash langle\; \backslash ,g\backslash ,\; |\; \backslash ,h\backslash ,\; \backslash rangle$ for every $h\; \backslash in\; H.$ The uniqueness of the (non-zero) vector $f\_\{\backslash varphi\}$ representing $\backslash varphi$ implies that $f\_\{\backslash varphi\}\; =\; g,$ which in turn implies that $\backslash overline\{\backslash varphi\; q\}\; \backslash neq\; 0$ and $q\; =\; \backslash frac\{\backslash |q\backslash |^2\}\{\backslash overline\{\backslash varphi\; q\}\}\; f\_\{\backslash varphi\}.$ Thus every vector in $K^\{\backslash bot\}$ is a scalar multiple of $f\_\{\backslash varphi\}.$ $\backslash blacksquare$

The formulas for the inner products follow from the polarization identity.

{{collapse bottom}}

If $\backslash varphi\; \backslash in\; H^*$ then

$$\backslash varphi\; \backslash left(f\_\{\backslash varphi\}\backslash right)\; =\; \backslash left\backslash langle\; f\_\{\backslash varphi\},\; f\_\{\backslash varphi\}\; \backslash right\backslash rangle\; =\; \backslash left\backslash |f\_\{\backslash varphi\}\backslash right\backslash |^2\; =\; \backslash |\backslash varphi\backslash |^2.$$

So in particular, $\backslash varphi\; \backslash left(f\_\{\backslash varphi\}\backslash right)\; \backslash geq\; 0$ is always real and furthermore, $\backslash varphi\; \backslash left(f\_\{\backslash varphi\}\backslash right)\; =\; 0$ if and only if $f\_\{\backslash varphi\}\; =\; 0$ if and only if $\backslash varphi\; =\; 0.$

Linear functionals as affine hyperplanes

A non-trivial continuous linear functional $\backslash varphi$ is often interpreted geometrically by identifying it with the affine hyperplane $A\; :=\; \backslash varphi^\{-1\}(1)$ (the kernel $\backslash ker\backslash varphi\; =\; \backslash varphi^\{-1\}(0)$ is also often visualized alongside $A\; :=\; \backslash varphi^\{-1\}(1)$ although knowing $A$ is enough to reconstruct $\backslash ker\; \backslash varphi$ because if $A\; =\; \backslash varnothing$ then $\backslash ker\; \backslash varphi\; =\; H$ and otherwise $\backslash ker\; \backslash varphi\; =\; A\; -\; A$). In particular, the norm of $\backslash varphi$ should somehow be interpretable as the "norm of the hyperplane $A$". When $\backslash varphi\; \backslash neq\; 0$ then the Riesz representation theorem provides such an interpretation of $\backslash |\backslash varphi\backslash |$ in terms of the affine hyperplane

$A\; :=\; \backslash varphi^\{-1\}(1)$ as follows: using the notation from the theorem's statement, from $\backslash |\backslash varphi\backslash |^2\; \backslash neq\; 0$ it follows that $C\; :=\; \backslash varphi^\{-1\}\backslash left(\backslash |\backslash varphi\backslash |^2\backslash right)\; =\; \backslash |\backslash varphi\backslash |^2\; \backslash varphi^\{-1\}(1)\; =\; \backslash |\backslash varphi\backslash |^2\; A$ and so $\backslash |\backslash varphi\backslash |\; =\; \backslash left\backslash |f\_\{\backslash varphi\}\backslash right\backslash |\; =\; \backslash inf\_\{c\; \backslash in\; C\}\; \backslash |c\backslash |$ implies $\backslash |\backslash varphi\backslash |\; =\; \backslash inf\_\{a\; \backslash in\; A\}\; \backslash |\backslash varphi\backslash |^2\; \backslash |a\backslash |$ and thus $\backslash |\backslash varphi\backslash |\; =\; \backslash frac\{1\}\{\backslash inf\_\{a\; \backslash in\; A\}\; \backslash |a\backslash |\}.$

This can also be seen by applying the Hilbert projection theorem to $A$ and concluding that the global minimum point of the map $A\; \backslash to\; [0,\; \backslash infty)$ defined by $a\; \backslash mapsto\; \backslash |a\backslash |$ is $\backslash frac\{f\_\{\backslash varphi\}\}\{\backslash |\backslash varphi\backslash |^2\}\; \backslash in\; A.$

The formulas

$$\backslash frac\{1\}\{\backslash inf\_\{a\; \backslash in\; A\}\; \backslash |a\backslash |\}\; =\; \backslash sup\_\{a\; \backslash in\; A\}\; \backslash frac\{1\}\{\backslash |a\backslash |\}$$

provide the promised interpretation of the linear functional's norm $\backslash |\backslash varphi\backslash |$ entirely in terms of its associated affine hyperplane $A\; =\; \backslash varphi^\{-1\}(1)$ (because with this formula, knowing only the {{em|set}} $A$ is enough to describe the norm of its associated linear {{em|functional}}). Defining $\backslash frac\{1\}\{\backslash infty\}\; :=\; 0,$ the infimum formula

$$\backslash |\backslash varphi\backslash |\; =\; \backslash frac\{1\}\{\backslash inf\_\{a\; \backslash in\; \backslash varphi^\{-1\}(1)\}\; \backslash |a\backslash |\}$$

will also hold when $\backslash varphi\; =\; 0.$

When the supremum is taken in $\backslash R$ (as is typically assumed), then the supremum of the empty set is $\backslash sup\; \backslash varnothing\; =\; -\; \backslash infty$ but if the supremum is taken in the non-negative reals $[0,\; \backslash infty)$ (which is the image/range of the norm $\backslash |\backslash ,\backslash cdot\backslash ,\backslash |$ when $\backslash dim\; H\; >\; 0$) then this supremum is instead $\backslash sup\; \backslash varnothing\; =\; 0,$ in which case the supremum formula $\backslash |\backslash varphi\backslash |\; =\; \backslash sup\_\{a\; \backslash in\; \backslash varphi^\{-1\}(1)\}\; \backslash frac\{1\}\{\backslash |a\backslash |\}$ will also hold when $\backslash varphi\; =\; 0$ (although the atypical equality $\backslash sup\; \backslash varnothing\; =\; 0$ is usually unexpected and so risks causing confusion).

Using the notation from the theorem above, several ways of constructing $f\_\{\backslash varphi\}$ from $\backslash varphi\; \backslash in\; H^*$ are now described.

If $\backslash varphi\; =\; 0$ then $f\_\{\backslash varphi\}\; :=\; 0$; in other words,

$$f\_0\; =\; 0.$$

This special case of $\backslash varphi\; =\; 0$ is henceforth assumed to be known, which is why some of the constructions given below start by assuming $\backslash varphi\; \backslash neq\; 0.$

Orthogonal complement of kernel

If $\backslash varphi\; \backslash neq\; 0$ then for any $0\; \backslash neq\; u\; \backslash in\; (\backslash ker\backslash varphi)^\{\backslash bot\},$

$$f\_\{\backslash varphi\}\; :=\; \backslash frac\{\backslash overline\{\backslash varphi(u)\}\; u\}\{\backslash |u\backslash |^2\}.$$

If $u\; \backslash in\; (\backslash ker\backslash varphi)^\{\backslash bot\}$ is a unit vector (meaning $\backslash |u\backslash |\; =\; 1$) then

$$f\_\{\backslash varphi\}\; :=\; \backslash overline\{\backslash varphi(u)\}\; u$$

(this is true even if $\backslash varphi\; =\; 0$ because in this case $f\_\{\backslash varphi\}\; =\; \backslash overline\{\backslash varphi(u)\}\; u\; =\; \backslash overline\{0\}\; u\; =\; 0$).

If $u$ is a unit vector satisfying the above condition then the same is true of $-u,$ which is also a unit vector in $(\backslash ker\backslash varphi)^\{\backslash bot\}.$ However, $\backslash overline\{\backslash varphi(-u)\}\; (-u)\; =\; \backslash overline\{\backslash varphi(u)\}\; u\; =\; f\_\backslash varphi$ so both these vectors result in the same $f\_\{\backslash varphi\}.$

Orthogonal projection onto kernel

If $x\; \backslash in\; H$ is such that $\backslash varphi(x)\; \backslash neq\; 0$ and if $x\_K$ is the orthogonal projection of $x$ onto $\backslash ker\backslash varphi$ then

$$f\_\{\backslash varphi\}\; =\; \backslash frac\{\backslash |\backslash varphi\backslash |^2\}\{\backslash varphi(x)\}\; \backslash left(x\; -\; x\_K\backslash right).$$

Orthonormal basis

Given an orthonormal basis $\backslash left\backslash \{e\_i\backslash right\backslash \}\_\{i\; \backslash in\; I\}$ of $H$ and a continuous linear functional $\backslash varphi\; \backslash in\; H^*,$ the vector $f\_\{\backslash varphi\}\; \backslash in\; H$ can be constructed uniquely by

$$f\_\backslash varphi\; =\; \backslash sum\_\{i\; \backslash in\; I\}\; \backslash overline\{\backslash varphi\backslash left(e\_i\backslash right)\}\; e\_i$$

where all but at most countably many $\backslash varphi\backslash left(e\_i\backslash right)$ will be equal to $0$ and where the value of $f\_\{\backslash varphi\}$ does not actually depend on choice of orthonormal basis (that is, using any other orthonormal basis for $H$ will result in the same vector).

If $y\; \backslash in\; H$ is written as $y\; =\; \backslash sum\_\{i\; \backslash in\; I\}\; a\_i\; e\_i$ then

$$\backslash varphi(y)\; =\; \backslash sum\_\{i\; \backslash in\; I\}\; \backslash varphi\backslash left(e\_i\backslash right)\; a\_i\; =\; \backslash langle\; f\_\{\backslash varphi\}\; |\; y\; \backslash rangle$$

and

$$\backslash left\backslash |f\_\{\backslash varphi\}\backslash right\backslash |^2\; =\; \backslash varphi\backslash left(f\_\{\backslash varphi\}\backslash right)\; =\; \backslash sum\_\{i\; \backslash in\; I\}\; \backslash varphi\backslash left(e\_i\backslash right)\; \backslash overline\{\backslash varphi\backslash left(e\_i\backslash right)\}\; =\; \backslash sum\_\{i\; \backslash in\; I\}\; \backslash left|\backslash varphi\backslash left(e\_i\backslash right)\backslash right|^2\; =\; \backslash |\backslash varphi\backslash |^2.$$

If the orthonormal basis $\backslash left\backslash \{e\_i\backslash right\backslash \}\_\{i\; \backslash in\; I\}\; =\; \backslash left\backslash \{e\_i\backslash right\backslash \}\_\{i=1\}^\{\backslash infty\}$ is a sequence then this becomes

$$f\_\backslash varphi\; =\; \backslash overline\{\backslash varphi\backslash left(e\_1\backslash right)\}\; e\_1\; +\; \backslash overline\{\backslash varphi\backslash left(e\_2\backslash right)\}\; e\_2\; +\; \backslash cdots$$

and if $y\; \backslash in\; H$ is written as $y\; =\; \backslash sum\_\{i\; \backslash in\; I\}\; a\_i\; e\_i\; =\; a\_1\; e\_1\; +\; a\_2\; e\_2\; +\; \backslash cdots$ then

$$\backslash varphi(y)\; =\; \backslash varphi\backslash left(e\_1\backslash right)\; a\_1\; +\; \backslash varphi\backslash left(e\_2\backslash right)\; a\_2\; +\; \backslash cdots\; =\; \backslash langle\; f\_\{\backslash varphi\}\; |\; y\; \backslash rangle.$$

Consider the special case of $H\; =\; \backslash Complex^n$ (where $n\; >\; 0$ is an integer) with the standard inner product

$$\backslash langle\; z\; \backslash mid\; w\; \backslash rangle\; :=\; \backslash overline\{\backslash ,\backslash vec\{z\}\backslash ,\backslash ,\}^\{\backslash operatorname\{T\}\}\; \backslash vec\{w\}\; \backslash qquad\; \backslash text\{\; for\; all\; \}\; \backslash ;\; w,\; z\; \backslash in\; H$$

where $w\; \backslash text\{\; and\; \}\; z$ are represented as column matrices $\backslash vec\{w\}\; :=\; \backslash begin\{bmatrix\}w\_1\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; w\_n\backslash end\{bmatrix\}$ and $\backslash vec\{z\}\; :=\; \backslash begin\{bmatrix\}z\_1\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; z\_n\backslash end\{bmatrix\}$ with respect to the standard orthonormal basis $e\_1,\; \backslash ldots,\; e\_n$ on $H$ (here, $e\_i$ is $1$ at its $i$^th coordinate and $0$ everywhere else; as usual, $H^*$ will now be associated with the dual basis) and where $\backslash overline\{\backslash ,\backslash vec\{z\}\backslash ,\}^\{\backslash operatorname\{T\}\}\; :=\; \backslash left[\backslash overline\{z\_1\},\; \backslash ldots,\; \backslash overline\{z\_n\}\backslash right]$ denotes the conjugate transpose of $\backslash vec\{z\}.$

Let $\backslash varphi\; \backslash in\; H^*$ be any linear functional and let $\backslash varphi\_1,\; \backslash ldots,\; \backslash varphi\_n\; \backslash in\; \backslash Complex$ be the unique scalars such that

$$\backslash varphi\backslash left(w\_1,\; \backslash ldots,\; w\_n\backslash right)\; =\; \backslash varphi\_1\; w\_1\; +\; \backslash cdots\; +\; \backslash varphi\_n\; w\_n\; \backslash qquad\; \backslash text\{\; for\; all\; \}\; \backslash ;\; w\; :=\; \backslash left(w\_1,\; \backslash ldots,\; w\_n\backslash right)\; \backslash in\; H,$$

where it can be shown that $\backslash varphi\_i\; =\; \backslash varphi\backslash left(e\_i\backslash right)$ for all $i\; =\; 1,\; \backslash ldots,\; n.$

Then the Riesz representation of $\backslash varphi$ is the vector

$$f\_\{\backslash varphi\}\; ~:=~\; \backslash overline\{\backslash varphi\_1\}\; e\_1\; +\; \backslash cdots\; +\; \backslash overline\{\backslash varphi\_n\}\; e\_n$$

~=~ \left(\overline{\varphi_1}, \ldots, \overline{\varphi_n}\right)

\in H.

To see why, identify every vector $w\; =\; \backslash left(w\_1,\; \backslash ldots,\; w\_n\backslash right)$ in $H$ with the column matrix

$\backslash vec\{w\}\; :=\; \backslash begin\{bmatrix\}w\_1\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; w\_n\backslash end\{bmatrix\}$

so that $f\_\{\backslash varphi\}$ is identified with $\backslash vec\{f\_\{\backslash varphi\}\}\; :=\; \backslash begin\{bmatrix\}\backslash overline\{\backslash varphi\_1\}\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; \backslash overline\{\backslash varphi\_n\}\backslash end\{bmatrix\}\; =\; \backslash begin\{bmatrix\}\backslash overline\{\backslash varphi\backslash left(e\_1\backslash right)\}\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; \backslash overline\{\backslash varphi\backslash left(e\_n\backslash right)\}\backslash end\{bmatrix\}.$

As usual, also identify the linear functional $\backslash varphi$ with its transformation matrix, which is the row matrix $\backslash vec\{\backslash varphi\}\; :=\; \backslash left[\backslash varphi\_1,\; \backslash ldots,\; \backslash varphi\_n\backslash right]$ so that $\backslash vec\{f\_\{\backslash varphi\}\}\; :=\; \backslash overline\{\backslash ,\backslash vec\{\backslash varphi\}\backslash ,\backslash ,\}^\{\backslash operatorname\{T\}\}$ and the function $\backslash varphi$ is the assignment $\backslash vec\{w\}\; \backslash mapsto\; \backslash vec\{\backslash varphi\}\; \backslash ,\; \backslash vec\{w\},$ where the right hand side is matrix multiplication. Then for all $w\; =\; \backslash left(w\_1,\; \backslash ldots,\; w\_n\backslash right)\; \backslash in\; H,$

$$\backslash varphi(w)$$

= \varphi_1 w_1 + \cdots + \varphi_n w_n

= \left[\varphi_1, \ldots, \varphi_n\right]

\begin{bmatrix}w_1 \\ \vdots \\ w_n\end{bmatrix}

= \overline{\begin{bmatrix}\overline{\varphi_1} \\ \vdots \\ \overline{\varphi_n}\end{bmatrix}}^{\operatorname{T}}

\vec{w}

= \overline{\,\vec{f_{\varphi}}\,\,}^{\operatorname{T}} \vec{w}

= \left\langle \,\,f_{\varphi}\, \mid \,w\, \right\rangle,

which shows that $f\_\{\backslash varphi\}$ satisfies the defining condition of the Riesz representation of $\backslash varphi.$

The bijective antilinear isometry $\backslash Phi\; :\; H\; \backslash to\; H^*$ defined in the corollary to the Riesz representation theorem is the assignment that sends $z\; =\; \backslash left(z\_1,\; \backslash ldots,\; z\_n\backslash right)\; \backslash in\; H$ to the linear functional $\backslash Phi(z)\; \backslash in\; H^*$ on $H$ defined by

$$w\; =\; \backslash left(w\_1,\; \backslash ldots,\; w\_n\backslash right)\; ~\backslash mapsto~\; \backslash langle\; \backslash ,z\backslash ,\; \backslash mid\; \backslash ,w\backslash ,\backslash rangle\; =\; \backslash overline\{z\_1\}\; w\_1\; +\; \backslash cdots\; +\; \backslash overline\{z\_n\}\; w\_n,$$

where under the identification of vectors in $H$ with column matrices and vector in $H^*$ with row matrices, $\backslash Phi$ is just the assignment

$$\backslash vec\{z\}\; =\; \backslash begin\{bmatrix\}z\_1\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; z\_n\backslash end\{bmatrix\}\; ~\backslash mapsto~\; \backslash overline\{\backslash ,\backslash vec\{z\}\backslash ,\}^\{\backslash operatorname\{T\}\}\; =\; \backslash left[\backslash overline\{z\_1\},\; \backslash ldots,\; \backslash overline\{z\_n\}\backslash right].$$

As described in the corollary, $\backslash Phi$'s inverse $\backslash Phi^\{-1\}\; :\; H^*\; \backslash to\; H$ is the antilinear isometry $\backslash varphi\; \backslash mapsto\; f\_\{\backslash varphi\},$ which was just shown above to be:

$$\backslash varphi\; ~\backslash mapsto~\; f\_\{\backslash varphi\}\; ~:=~\; \backslash left(\backslash overline\{\backslash varphi\backslash left(e\_1\backslash right)\},\; \backslash ldots,\; \backslash overline\{\backslash varphi\backslash left(e\_n\backslash right)\}\backslash right);$$

where in terms of matrices, $\backslash Phi^\{-1\}$ is the assignment

$$\backslash vec\{\backslash varphi\}\; =\; \backslash left[\backslash varphi\_1,\; \backslash ldots,\; \backslash varphi\_n\backslash right]\; ~\backslash mapsto~\; \backslash overline\{\backslash ,\backslash vec\{\backslash varphi\}\backslash ,\backslash ,\}^\{\backslash operatorname\{T\}\}\; =\; \backslash begin\{bmatrix\}\backslash overline\{\backslash varphi\_1\}\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; \backslash overline\{\backslash varphi\_n\}\backslash end\{bmatrix\}.$$

Thus in terms of matrices, each of $\backslash Phi\; :\; H\; \backslash to\; H^*$ and $\backslash Phi^\{-1\}\; :\; H^*\; \backslash to\; H$ is just the operation of conjugate transposition $\backslash vec\{v\}\; \backslash mapsto\; \backslash overline\{\backslash ,\backslash vec\{v\}\backslash ,\}^\{\backslash operatorname\{T\}\}$ (although between different spaces of matrices: if $H$ is identified with the space of all column (respectively, row) matrices then $H^*$ is identified with the space of all row (respectively, column matrices).

This example used the standard inner product, which is the map $\backslash langle\; z\; \backslash mid\; w\; \backslash rangle\; :=\; \backslash overline\{\backslash ,\backslash vec\{z\}\backslash ,\backslash ,\}^\{\backslash operatorname\{T\}\}\; \backslash vec\{w\},$ but if a different inner product is used, such as $\backslash langle\; z\; \backslash mid\; w\; \backslash rangle\_M\; :=\; \backslash overline\{\backslash ,\backslash vec\{z\}\backslash ,\backslash ,\}^\{\backslash operatorname\{T\}\}\; \backslash ,\; M\; \backslash ,\; \backslash vec\{w\}\; \backslash ,$ where $M$ is any Hermitian positive-definite matrix, or if a different orthonormal basis is used then the transformation matrices, and thus also the above formulas, will be different.

{{See also|Complexification}}

Assume that $H$ is a complex Hilbert space with inner product $\backslash langle\; \backslash ,\backslash cdot\backslash mid\backslash cdot\backslash ,\; \backslash rangle.$

When the Hilbert space $H$ is reinterpreted as a real Hilbert space then it will be denoted by $H\_\{\backslash R\},$ where the (real) inner-product on $H\_\{\backslash R\}$ is the real part of $H$'s inner product; that is:

$$\backslash langle\; x,\; y\; \backslash rangle\_\{\backslash R\}\; :=\; \backslash operatorname\{re\}\; \backslash langle\; x,\; y\; \backslash rangle.$$

The norm on $H\_\{\backslash R\}$ induced by $\backslash langle\; \backslash ,\backslash cdot\backslash ,,\; \backslash ,\backslash cdot\backslash ,\; \backslash rangle\_\{\backslash R\}$ is equal to the original norm on $H$ and the continuous dual space of $H\_\{\backslash R\}$ is the set of all {{em|real}}-valued bounded $\backslash R$-linear functionals on $H\_\{\backslash R\}$ (see the article about the polarization identity for additional details about this relationship).

Let $\backslash psi\_\{\backslash R\}\; :=\; \backslash operatorname\{re\}\; \backslash psi$ and $\backslash psi\_\{i\}\; :=\; \backslash operatorname\{im\}\; \backslash psi$ denote the real and imaginary parts of a linear functional $\backslash psi,$ so that $\backslash psi\; =\; \backslash operatorname\{re\}\; \backslash psi\; +\; i\; \backslash operatorname\{im\}\; \backslash psi\; =\; \backslash psi\_\{\backslash R\}\; +\; i\; \backslash psi\_\{i\}.$

The formula expressing a linear functional in terms of its real part is

$$\backslash psi(h)\; =\; \backslash psi\_\{\backslash R\}(h)\; -\; i\; \backslash psi\_\{\backslash R\}\; (i\; h)\; \backslash quad\; \backslash text\{\; for\; \}\; h\; \backslash in\; H,$$

where $\backslash psi\_\{i\}(h)\; =\; -\; i\; \backslash psi\_\{\backslash R\}\; (i\; h)$ for all $h\; \backslash in\; H.$

It follows that $\backslash ker\backslash psi\_\{\backslash R\}\; =\; \backslash psi^\{-1\}(i\; \backslash R),$ and that $\backslash psi\; =\; 0$ if and only if $\backslash psi\_\{\backslash R\}\; =\; 0.$

It can also be shown that $\backslash |\backslash psi\backslash |\; =\; \backslash left\backslash |\backslash psi\_\{\backslash R\}\backslash right\backslash |\; =\; \backslash left\backslash |\backslash psi\_i\backslash right\backslash |$ where $\backslash left\backslash |\backslash psi\_\{\backslash R\}\backslash right\backslash |\; :=\; \backslash sup\_\{\backslash |h\backslash |\; \backslash leq\; 1\}\; \backslash left|\backslash psi\_\{\backslash R\}(h)\backslash right|$ and $\backslash left\backslash |\backslash psi\_i\backslash right\backslash |\; :=\; \backslash sup\_\{\backslash |h\backslash |\; \backslash leq\; 1\}\; \backslash left|\backslash psi\_i(h)\backslash right|$ are the usual operator norms.

In particular, a linear functional $\backslash psi$ is bounded if and only if its real part $\backslash psi\_\{\backslash R\}$ is bounded.

Representing a functional and its real part

The Riesz representation of a continuous linear function $\backslash varphi$ on a complex Hilbert space is equal to the Riesz representation of its real part $\backslash operatorname\{re\}\; \backslash varphi$ on its associated real Hilbert space.

Explicitly, let $\backslash varphi\; \backslash in\; H^*$ and as above, let $f\_\backslash varphi\; \backslash in\; H$ be the Riesz representation of $\backslash varphi$ obtained in $(H,\; \backslash langle,\; \backslash cdot,\; \backslash cdot\; \backslash rangle),$ so it is the unique vector that satisfies $\backslash varphi(x)\; =\; \backslash left\backslash langle\; f\_\{\backslash varphi\}\; \backslash mid\; x\; \backslash right\backslash rangle$ for all $x\; \backslash in\; H.$

The real part of $\backslash varphi$ is a continuous real linear functional on $H\_\{\backslash R\}$ and so the Riesz representation theorem may be applied to $\backslash varphi\_\{\backslash R\}\; :=\; \backslash operatorname\{re\}\; \backslash varphi$ and the associated real Hilbert space $\backslash left(H\_\{\backslash R\},\; \backslash langle,\; \backslash cdot,\; \backslash cdot\; \backslash rangle\_\{\backslash R\}\backslash right)$ to produce its Riesz representation, which will be denoted by $f\_\{\backslash varphi\_\{\backslash R\}\}.$

That is, $f\_\{\backslash varphi\_\{\backslash R\}\}$ is the unique vector in $H\_\{\backslash R\}$ that satisfies $\backslash varphi\_\{\backslash R\}(x)\; =\; \backslash left\backslash langle\; f\_\{\backslash varphi\_\{\backslash R\}\}\; \backslash mid\; x\; \backslash right\backslash rangle\_\{\backslash R\}$ for all $x\; \backslash in\; H.$

The conclusion is $f\_\{\backslash varphi\_\{\backslash R\}\}\; =\; f\_\{\backslash varphi\}.$

This follows from the main theorem because $\backslash ker\backslash varphi\_\{\backslash R\}\; =\; \backslash varphi^\{-1\}(i\; \backslash R)$ and if $x\; \backslash in\; H$ then

$$\backslash left\backslash langle\; f\_\backslash varphi\; \backslash mid\; x\; \backslash right\backslash rangle\_\{\backslash R\}\; =\; \backslash operatorname\{re\}\; \backslash left\backslash langle\; f\_\backslash varphi\; \backslash mid\; x\; \backslash right\backslash rangle\; =\; \backslash operatorname\{re\}\; \backslash varphi(x)\; =\; \backslash varphi\_\{\backslash R\}(x)$$

and consequently, if $m\; \backslash in\; \backslash ker\backslash varphi\_\{\backslash R\}$ then $\backslash left\backslash langle\; f\_\{\backslash varphi\}\backslash mid\; m\; \backslash right\backslash rangle\_\{\backslash R\}\; =\; 0,$ which shows that $f\_\{\backslash varphi\}\; \backslash in\; (\backslash ker\backslash varphi\_\{\backslash R\})^\{\backslash perp\_\{\backslash R\}\}.$

Moreover, $\backslash varphi(f\_\backslash varphi)\; =\; \backslash |\backslash varphi\backslash |^2$ being a real number implies that $\backslash varphi\_\{\backslash R\}\; (f\_\backslash varphi)\; =\; \backslash operatorname\{re\}\; \backslash varphi(f\_\backslash varphi)\; =\; \backslash |\backslash varphi\backslash |^2.$

In other words, in the theorem and constructions above, if $H$ is replaced with its real Hilbert space counterpart $H\_\{\backslash R\}$ and if $\backslash varphi$ is replaced with $\backslash operatorname\{re\}\; \backslash varphi$ then $f\_\{\backslash varphi\}\; =\; f\_\{\backslash operatorname\{re\}\; \backslash varphi\}.$ This means that vector $f\_\{\backslash varphi\}$ obtained by using $\backslash left(H\_\{\backslash R\},\; \backslash langle,\; \backslash cdot,\; \backslash cdot\; \backslash rangle\_\{\backslash R\}\backslash right)$ and the real linear functional $\backslash operatorname\{re\}\; \backslash varphi$ is the equal to the vector obtained by using the origin complex Hilbert space $\backslash left(H,\; \backslash left\backslash langle,\; \backslash cdot,\; \backslash cdot\; \backslash right\backslash rangle\backslash right)$ and original complex linear functional $\backslash varphi$ (with identical norm values as well).

Furthermore, if $\backslash varphi\; \backslash neq\; 0$ then $f\_\{\backslash varphi\}$ is perpendicular to $\backslash ker\backslash varphi\_\{\backslash R\}$ with respect to $\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle\_\{\backslash R\}$ where the kernel of $\backslash varphi$ is be a proper subspace of the kernel of its real part $\backslash varphi\_\{\backslash R\}.$ Assume now that $\backslash varphi\; \backslash neq\; 0.$

Then $f\_\{\backslash varphi\}\; \backslash not\backslash in\; \backslash ker\backslash varphi\_\{\backslash R\}$ because $\backslash varphi\_\{\backslash R\}\backslash left(f\_\{\backslash varphi\}\backslash right)\; =\; \backslash varphi\backslash left(f\_\{\backslash varphi\}\backslash right)\; =\; \backslash |\backslash varphi\backslash |^2\; \backslash neq\; 0$ and $\backslash ker\backslash varphi$ is a proper subset of $\backslash ker\backslash varphi\_\{\backslash R\}.$ The vector subspace $\backslash ker\; \backslash varphi$ has real codimension $1$ in $\backslash ker\backslash varphi\_\{\backslash R\},$ while $\backslash ker\backslash varphi\_\{\backslash R\}$ has {{em|real}} codimension $1$ in $H\_\{\backslash R\},$ and $\backslash left\backslash langle\; f\_\{\backslash varphi\},\; \backslash ker\backslash varphi\_\{\backslash R\}\; \backslash right\backslash rangle\_\{\backslash R\}\; =\; 0.$ That is, $f\_\{\backslash varphi\}$ is perpendicular to $\backslash ker\backslash varphi\_\{\backslash R\}$ with respect to $\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle\_\{\backslash R\}.$

Induced linear map into anti-dual

The map defined by placing $y$ into the {{em|linear}} coordinate of the inner product and letting the variable $h\; \backslash in\; H$ vary over the {{em|antilinear}} coordinate results in an Antilinear map:

$$\backslash langle\; \backslash ,\backslash cdot\; \backslash mid\; y\backslash ,\; \backslash rangle\; =\; \backslash langle\; \backslash ,y,\; \backslash cdot\backslash ,\; \backslash rangle\; :\; H\; \backslash to\; \backslash mathbb\{F\}\; \backslash quad\; \backslash text\{\; defined\; by\; \}\; \backslash quad\; h\; \backslash mapsto\; \backslash langle\; \backslash ,h\; \backslash mid\; y\backslash ,\; \backslash rangle\; =\; \backslash langle\; \backslash ,y,\; h\backslash ,\; \backslash rangle.$$

This map is an element of $\backslash overline\{H\}^*,$ which is the continuous anti-dual space of $H.$

The {{em|canonical map from $H$ into its anti-dual}} $\backslash overline\{H\}^*${{sfn|Trèves|2006|pp=112–123}} is the Linear operator

$$\backslash begin\{alignat\}\{4\}$$

\operatorname{In}_H^{\overline{H}^*} :\;&& H &&\;\to \;& \overline{H}^* \\[0.3ex]

&& y &&\;\mapsto\;& \langle \,\cdot \mid y\, \rangle = \langle \,y, \cdot\, \rangle \\[0.3ex]

\end{alignat}

which is also an injective isometry.{{sfn|Trèves|2006|pp=112–123}}

The Fundamental theorem of Hilbert spaces, which is related to Riesz representation theorem, states that this map is surjective (and thus bijective). Consequently, every antilinear functional on $H$ can be written (uniquely) in this form.{{sfn|Trèves|2006|pp=112–123}}

If $\backslash operatorname\{Cong\}\; :\; H^*\; \backslash to\; \backslash overline\{H\}^*$ is the canonical Antilinear map bijective isometry $f\; \backslash mapsto\; \backslash overline\{f\}$ that was defined above, then the following equality holds:

$$\backslash operatorname\{Cong\}\; ~\backslash circ~\; \backslash operatorname\{In\}\_H^\{H^*\}\; ~=~\; \backslash operatorname\{In\}\_H^\{\backslash overline\{H\}^*\}.$$

{{Main|Bra–ket notation}}

Let $\backslash left(H,\; \backslash langle\backslash cdot,\; \backslash cdot\; \backslash rangle\_H\backslash right)$ be a Hilbert space and as before, let $\backslash langle\; y\backslash ,\; |\; \backslash ,x\; \backslash rangle\_H\; :=\; \backslash langle\; x,\; y\; \backslash rangle\_H.$

Let

$$\backslash begin\{alignat\}\{4\}$$

\Phi :\;&& H &&\;\to \;& H^* \\[0.3ex]

&& g &&\;\mapsto\;& \left\langle \,g\mid \cdot\, \right\rangle_H = \left\langle \,\cdot, g\, \right\rangle_H \\

\end{alignat}

which is a bijective antilinear isometry that satisfies

$$(\backslash Phi\; h)\; g\; =\; \backslash langle\; h\backslash mid\; g\; \backslash rangle\_H\; =\; \backslash langle\; g,\; h\; \backslash rangle\_H\; \backslash quad\; \backslash text\{\; for\; all\; \}\; g,\; h\; \backslash in\; H.$$

Bras

Given a vector $h\; \backslash in\; H,$ let $\backslash langle\; h\backslash ,\; |$ denote the continuous linear functional $\backslash Phi\; h$; that is,

$$\backslash langle\; h\backslash ,\; |\; ~:=~\; \backslash Phi\; h$$

so that this functional $\backslash langle\; h\backslash ,\; |$ is defined by $g\; \backslash mapsto\; \backslash left\backslash langle\; \backslash ,h\backslash mid\; g\backslash ,\; \backslash right\backslash rangle\_H.$ This map was denoted by $\backslash left\backslash langle\; h\; \backslash mid\; \backslash cdot\backslash ,\; \backslash right\backslash rangle$ earlier in this article.

The assignment $h\; \backslash mapsto\; \backslash langle\; h\; |$ is just the isometric antilinear isomorphism $\backslash Phi\; ~:~\; H\; \backslash to\; H^*,$ which is why $~\backslash langle\; c\; g\; +\; h\backslash ,\; |\; ~=~\; \backslash overline\{c\}\; \backslash langle\; g\backslash mid\; ~+~\; \backslash langle\; h\backslash ,\; |~$ holds for all $g,\; h\; \backslash in\; H$ and all scalars $c.$

The result of plugging some given $g\; \backslash in\; H$ into the functional $\backslash langle\; h\backslash ,\; |$ is the scalar $\backslash langle\; h\backslash ,\; |\; \backslash ,g\; \backslash rangle\_H\; =\; \backslash langle\; g,\; h\; \backslash rangle\_H,$ which may be denoted by $\backslash langle\; h\; \backslash mid\; g\; \backslash rangle.$The usual notation for plugging an element $g$ into a linear map $F$ is $F(g)$ and sometimes $Fg.$ Replacing $F$ with $\backslash langle\; h\backslash mid\; :=~\; \backslash Phi\; h$ produces $\backslash langle\; h\backslash mid(g)$ or $\backslash langle\; h\; \backslash mid\; g,$ which is unsightly (despite being consistent with the usual notation used with functions). Consequently, the symbol $\backslash ,\backslash rangle\backslash ,$ is appended to the end, so that the notation $\backslash langle\; h\backslash mid\; g\; \backslash rangle$ is used instead to denote this value $(\backslash Phi\; h)\; g.$

Bra of a linear functional

Given a continuous linear functional $\backslash psi\; \backslash in\; H^*,$ let $\backslash langle\; \backslash psi\backslash mid$ denote the vector $\backslash Phi^\{-1\}\; \backslash psi\; \backslash in\; H$; that is,

$$\backslash langle\; \backslash psi\backslash mid\; ~:=~\; \backslash Phi^\{-1\}\; \backslash psi.$$

The assignment $\backslash psi\; \backslash mapsto\; \backslash langle\; \backslash psi\backslash mid$ is just the isometric antilinear isomorphism $\backslash Phi^\{-1\}\; ~:~\; H^*\; \backslash to\; H,$ which is why $~\backslash langle\; c\; \backslash psi\; +\; \backslash phi\backslash mid\; ~=~\; \backslash overline\{c\}\; \backslash langle\; \backslash psi\backslash mid\; ~+~\; \backslash langle\; \backslash phi\backslash mid~$ holds for all $\backslash phi,\; \backslash psi\; \backslash in\; H^*$ and all scalars $c.$

The defining condition of the vector $\backslash langle\; \backslash psi\; |\; \backslash in\; H$ is the technically correct but unsightly equality

$$\backslash left\backslash langle\; \backslash ,\; \backslash langle\; \backslash psi\backslash mid\; \backslash ,\; \backslash mid\; g\; \backslash right\backslash rangle\_H\; ~=~\; \backslash psi\; g\; \backslash quad\; \backslash text\{\; for\; all\; \}\; g\; \backslash in\; H,$$

which is why the notation $\backslash left\backslash langle\; \backslash psi\; \backslash mid\; g\; \backslash right\backslash rangle$ is used in place of $\backslash left\backslash langle\; \backslash ,\; \backslash langle\; \backslash psi\backslash mid\; \backslash ,\; \backslash mid\; g\; \backslash right\backslash rangle\_H\; =\; \backslash left\backslash langle\; g,\; \backslash ,\; \backslash langle\; \backslash psi\backslash mid\; \backslash right\backslash rangle\_H.$ With this notation, the defining condition becomes

$$\backslash left\backslash langle\; \backslash psi\backslash mid\; g\; \backslash right\backslash rangle\; ~=~\; \backslash psi\; g\; \backslash quad\; \backslash text\{\; for\; all\; \}\; g\; \backslash in\; H.$$

Kets

For any given vector $g\; \backslash in\; H,$ the notation $|\; \backslash ,g\; \backslash rangle$ is used to denote $g$; that is,

$$\backslash mid\; g\; \backslash rangle\; :\; =\; g.$$

The assignment $g\; \backslash mapsto\; |\; \backslash ,g\; \backslash rangle$ is just the identity map $\backslash operatorname\{Id\}\_H\; :\; H\; \backslash to\; H,$ which is why $~\backslash mid\; c\; g\; +\; h\; \backslash rangle\; ~=~\; c\; \backslash mid\; g\; \backslash rangle\; ~+~\; \backslash mid\; h\; \backslash rangle~$ holds for all $g,\; h\; \backslash in\; H$ and all scalars $c.$

The notation $\backslash langle\; h\backslash mid\; g\; \backslash rangle$ and $\backslash langle\; \backslash psi\backslash mid\; g\; \backslash rangle$ is used in place of $\backslash left\backslash langle\; h\backslash mid\; \backslash ,\; \backslash mid\; g\; \backslash rangle\; \backslash ,\; \backslash right\backslash rangle\_H\; ~=~\; \backslash left\backslash langle\; \backslash mid\; g\; \backslash rangle,\; h\; \backslash right\backslash rangle\_H$ and $\backslash left\backslash langle\; \backslash psi\backslash mid\; \backslash ,\; \backslash mid\; g\; \backslash rangle\; \backslash ,\; \backslash right\backslash rangle\_H\; ~=~\; \backslash left\backslash langle\; g,\; \backslash ,\; \backslash langle\; \backslash psi\backslash mid\; \backslash right\backslash rangle\_H,$ respectively. As expected, $~\backslash langle\; \backslash psi\backslash mid\; g\; \backslash rangle\; =\; \backslash psi\; g~$ and $~\backslash langle\; h\backslash mid\; g\; \backslash rangle~$ really is just the scalar $~\backslash langle\; h\backslash mid\; g\; \backslash rangle\_H\; ~=~\; \backslash langle\; g,\; h\; \backslash rangle\_H.$

Let $A\; :\; H\; \backslash to\; Z$ be a continuous linear operator between Hilbert spaces $\backslash left(H,\; \backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle\_H\backslash right)$ and $\backslash left(Z,\; \backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle\_Z\; \backslash right).$ As before, let $\backslash langle\; y\; \backslash mid\; x\; \backslash rangle\_H\; :=\; \backslash langle\; x,\; y\; \backslash rangle\_H$ and $\backslash langle\; y\; \backslash mid\; x\; \backslash rangle\_Z\; :=\; \backslash langle\; x,\; y\; \backslash rangle\_Z.$

Denote by

$$\backslash begin\{alignat\}\{4\}$$

\Phi_H :\;&& H &&\;\to \;& H^* \\[0.3ex]

&& g &&\;\mapsto\;& \langle \,g \mid \cdot\, \rangle_H \\

\end{alignat}

\quad \text{ and } \quad

\begin{alignat}{4}

\Phi_Z :\;&& Z &&\;\to \;& Z^* \\[0.3ex]

&& y &&\;\mapsto\;& \langle \,y \mid \cdot\, \rangle_Z \\

\end{alignat}

the usual bijective antilinear isometries that satisfy:

$$\backslash left(\backslash Phi\_H\; g\backslash right)\; h\; =\; \backslash langle\; g\backslash mid\; h\; \backslash rangle\_H\; \backslash quad\; \backslash text\{\; for\; all\; \}\; g,\; h\; \backslash in\; H\; \backslash qquad\; \backslash text\{\; and\; \}\; \backslash qquad\; \backslash left(\backslash Phi\_Z\; y\backslash right)\; z\; =\; \backslash langle\; y\; \backslash mid\; z\; \backslash rangle\_Z\; \backslash quad\; \backslash text\{\; for\; all\; \}\; y,\; z\; \backslash in\; Z.$$

{{Main|Hermitian adjoint|Conjugate transpose}}

For every $z\; \backslash in\; Z,$ the scalar-valued map $\backslash langle\; z\backslash mid\; A\; (\backslash cdot)\; \backslash rangle\_Z$ on $H$ defined by

$$h\; \backslash mapsto\; \backslash langle\; z\backslash mid\; A\; h\; \backslash rangle\_Z\; =\; \backslash langle\; A\; h,\; z\; \backslash rangle\_Z$$

is a continuous linear functional on $H$ and so by the Riesz representation theorem, there exists a unique vector in $H,$ denoted by $A^*\; z,$ such that $\backslash langle\; z\; \backslash mid\; A\; (\backslash cdot)\; \backslash rangle\_Z\; =\; \backslash left\backslash langle\; A^*\; z\; \backslash mid\; \backslash cdot\backslash ,\; \backslash right\backslash rangle\_H,$ or equivalently, such that

$$\backslash langle\; z\; \backslash mid\; A\; h\; \backslash rangle\_Z\; =\; \backslash left\backslash langle\; A^*\; z\; \backslash mid\; h\; \backslash right\backslash rangle\_H\; \backslash quad\; \backslash text\{\; for\; all\; \}\; h\; \backslash in\; H.$$

The assignment $z\; \backslash mapsto\; A^*\; z$ thus induces a function $A^*\; :\; Z\; \backslash to\; H$ called the {{em|adjoint}} of $A\; :\; H\; \backslash to\; Z$ whose defining condition is

$$\backslash langle\; z\; \backslash mid\; A\; h\; \backslash rangle\_Z\; =\; \backslash left\backslash langle\; A^*\; z\backslash mid\; h\; \backslash right\backslash rangle\_H\; \backslash quad\; \backslash text\{\; for\; all\; \}\; h\; \backslash in\; H\; \backslash text\{\; and\; all\; \}\; z\; \backslash in\; Z.$$

The adjoint $A^*\; :\; Z\; \backslash to\; H$ is necessarily a continuous (equivalently, a bounded) linear operator.

If $H$ is finite dimensional with the standard inner product and if $M$ is the transformation matrix of $A$ with respect to the standard orthonormal basis then $M$'s conjugate transpose $\backslash overline\{M^\{\backslash operatorname\{T\}\}\}$ is the transformation matrix of the adjoint $A^*.$

{{Main|Transpose of a linear map}}

{{See also|Transpose}}

It is also possible to define the {{em|transpose}} or {{em|algebraic adjoint}} of $A\; :\; H\; \backslash to\; Z,$ which is the map $\{\}^\{t\}A\; :\; Z^*\; \backslash to\; H^*$ defined by sending a continuous linear functionals $\backslash psi\; \backslash in\; Z^*$ to

$$\{\}^\{t\}A(\backslash psi)\; :=\; \backslash psi\; \backslash circ\; A,$$

where the composition $\backslash psi\; \backslash circ\; A$ is always a continuous linear functional on $H$ and it satisfies $\backslash |A\backslash |\; =\; \backslash left\backslash |\{\}^t\; A\backslash right\backslash |$ (this is true more generally, when $H$ and $Z$ are merely normed spaces).{{sfn|Rudin|1991|pp=92-115}}

So for example, if $z\; \backslash in\; Z$ then $\{\}^\{t\}A$ sends the continuous linear functional $\backslash langle\; z\; \backslash mid\; \backslash cdot\; \backslash rangle\_Z\; \backslash in\; Z^*$ (defined on $Z$ by $g\; \backslash mapsto\; \backslash langle\; z\; \backslash mid\; g\; \backslash rangle\_Z$) to the continuous linear functional $\backslash langle\; z\; \backslash mid\; A(\backslash cdot)\; \backslash rangle\_Z\; \backslash in\; H^*$ (defined on $H$ by $h\; \backslash mapsto\; \backslash langle\; z\; \backslash mid\; A(h)\; \backslash rangle\_Z$);

using bra-ket notation, this can be written as $\{\}^\{t\}A\; \backslash langle\; z\; \backslash mid\; ~=~\; \backslash langle\; z\; \backslash mid\; A$ where the juxtaposition of $\backslash langle\; z\; \backslash mid$ with $A$ on the right hand side denotes function composition: $H\; \backslash xrightarrow\{A\}\; Z\; \backslash xrightarrow\{\backslash langle\; z\; \backslash mid\}\; \backslash Complex.$

The adjoint $A^*\; :\; Z\; \backslash to\; H$ is actually just to the transpose $\{\}^\{t\}A\; :\; Z^*\; \backslash to\; H^*${{sfn|Rudin|1991|pp=306-312}} when the Riesz representation theorem is used to identify $Z$ with $Z^*$ and $H$ with $H^*.$

Explicitly, the relationship between the adjoint and transpose is:

{{NumBlk|:|$\{\}^\{t\}A\; ~\backslash circ~\; \backslash Phi\_Z\; ~=~\; \backslash Phi\_H\; ~\backslash circ~\; A^*$|{{EquationRef|Adjoint-transpose}}|LnSty=1px dashed black}}

which can be rewritten as:

$$A^*\; ~=~\; \backslash Phi\_H^\{-1\}\; ~\backslash circ~\; \{\}^\{t\}A\; ~\backslash circ~\; \backslash Phi\_Z\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash quad\; \{\}^\{t\}A\; ~=~\; \backslash Phi\_H\; ~\backslash circ~\; A^*\; ~\backslash circ~\; \backslash Phi\_Z^\{-1\}.$$

{{Math proof|title=Proof|drop=hidden|proof=

To show that $\{\}^\{t\}A\; ~\backslash circ~\; \backslash Phi\_Z\; ~=~\; \backslash Phi\_H\; ~\backslash circ~\; A^*,$ fix $z\; \backslash in\; Z.$

The definition of $\{\}^\{t\}A$ implies $$\backslash left(\{\}^\{t\}A\; \backslash circ\; \backslash Phi\_Z\backslash right)\; z\; =\; \{\}^\{t\}A\; \backslash left(\backslash Phi\_Z\; z\backslash right)\; =\; \backslash left(\backslash Phi\_Z\; z\backslash right)\; \backslash circ\; A$$ so it remains to show that $\backslash left(\backslash Phi\_Z\; z\backslash right)\; \backslash circ\; A\; =\; \backslash Phi\_H\backslash left(A^*\; z\backslash right).$ If $h\; \backslash in\; H$ then $$\backslash left(\backslash left(\backslash Phi\_Z\; z\backslash right)\; \backslash circ\; A\backslash right)\; h\; =\; \backslash left(\backslash Phi\_Z\; z\backslash right)(A\; h)\; =\; \backslash langle\; z\backslash mid\; A\; h\; \backslash rangle\_Z\; =\; \backslash langle\; A^*\; z\backslash mid\; h\; \backslash rangle\_H\; =\; \backslash left(\backslash Phi\_H(A^*\; z)\backslash right)\; h,$$ as desired. $\backslash blacksquare$

}}

Alternatively, the value of the left and right hand sides of ({{EquationNote|Adjoint-transpose}}) at any given $z\; \backslash in\; Z$ can be rewritten in terms of the inner products as:

$$\backslash left(\{\}^\{t\}A\; ~\backslash circ~\; \backslash Phi\_Z\backslash right)\; z\; =\; \backslash langle\; z\; \backslash mid\; A\; (\backslash cdot)\; \backslash rangle\_Z\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash quad\backslash left(\backslash Phi\_H\; ~\backslash circ~\; A^*\backslash right)\; z\; =\; \backslash langle\; A^*\; z\backslash mid\backslash cdot\backslash ,\; \backslash rangle\_H$$

so that $\{\}^\{t\}A\; ~\backslash circ~\; \backslash Phi\_Z\; ~=~\; \backslash Phi\_H\; ~\backslash circ~\; A^*$ holds if and only if $\backslash langle\; z\; \backslash mid\; A\; (\backslash cdot)\; \backslash rangle\_Z\; =\; \backslash langle\; A^*\; z\backslash mid\backslash cdot\backslash ,\; \backslash rangle\_H$ holds; but the equality on the right holds by definition of $A^*\; z.$

The defining condition of $A^*\; z$ can also be written

$$\backslash langle\; z\; \backslash mid\; A\; ~=~\; \backslash langle\; A^*z\; \backslash mid$$

if bra-ket notation is used.

Assume $Z\; =\; H$ and let $\backslash Phi\; :=\; \backslash Phi\_H\; =\; \backslash Phi\_Z.$

Let $A\; :\; H\; \backslash to\; H$ be a continuous (that is, bounded) linear operator.

Whether or not $A\; :\; H\; \backslash to\; H$ is self-adjoint, normal, or unitary depends entirely on whether or not $A$ satisfies certain defining conditions related to its adjoint, which was shown by ({{EquationNote|Adjoint-transpose}}) to essentially be just the transpose $\{\}^t\; A\; :\; H^*\; \backslash to\; H^*.$

Because the transpose of $A$ is a map between continuous linear functionals, these defining conditions can consequently be re-expressed entirely in terms of linear functionals, as the remainder of subsection will now describe in detail.

The linear functionals that are involved are the simplest possible continuous linear functionals on $H$ that can be defined entirely in terms of $A,$ the inner product $\backslash langle\; \backslash ,\backslash cdot\backslash mid\backslash cdot\backslash ,\; \backslash rangle$ on $H,$ and some given vector $h\; \backslash in\; H.$

Specifically, these are $\backslash left\backslash langle\; A\; h\backslash mid\backslash cdot\backslash ,\; \backslash right\backslash rangle$ and $\backslash langle\; h\backslash mid\; A\; (\backslash cdot)\; \backslash rangle$ where

$$\backslash left\backslash langle\; A\; h\backslash mid\backslash cdot\backslash ,\; \backslash right\backslash rangle\; =\; \backslash Phi\; (A\; h)\; =\; (\backslash Phi\; \backslash circ\; A)\; h\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash quad\; \backslash langle\; h\backslash mid\; A\; (\backslash cdot)\; \backslash rangle\; =\; \backslash left(\{\}^\{t\}A\; \backslash circ\; \backslash Phi\backslash right)\; h.$$

Self-adjoint operators

{{See also|Self-adjoint operator|Hermitian matrix|Symmetric matrix}}

A continuous linear operator $A\; :\; H\; \backslash to\; H$ is called self-adjoint if it is equal to its own adjoint; that is, if $A\; =\; A^*.$ Using ({{EquationNote|Adjoint-transpose}}), this happens if and only if:

$$\backslash Phi\; \backslash circ\; A\; =\; \{\}^t\; A\; \backslash circ\; \backslash Phi$$

where this equality can be rewritten in the following two equivalent forms:

$$A\; =\; \backslash Phi^\{-1\}\; \backslash circ\; \{\}^t\; A\; \backslash circ\; \backslash Phi\; \backslash quad\; \backslash text\{\; or\; \}\; \backslash quad\; \{\}^\{t\}A\; =\; \backslash Phi\; \backslash circ\; A\; \backslash circ\; \backslash Phi^\{-1\}.$$

Unraveling notation and definitions produces the following characterization of self-adjoint operators in terms of the aforementioned continuous linear functionals: $A$ is self-adjoint if and only if for all $z\; \backslash in\; H,$ the linear functional $\backslash langle\; z\backslash mid\; A\; (\backslash cdot)\; \backslash rangle$ is equal to the linear functional $\backslash langle\; A\; z\backslash mid\backslash cdot\backslash ,\; \backslash rangle$; that is, if and only if

{{NumBlk|:|$\backslash langle\; z\backslash mid\; A\; (\backslash cdot)\; \backslash rangle\; =\; \backslash langle\; A\; z\backslash mid\; \backslash cdot\backslash ,\; \backslash rangle\; \backslash quad\; \backslash text\{\; for\; all\; \}\; z\; \backslash in\; H$|{{EquationRef|Self-adjointness functionals}}|LnSty=1px dashed black}}

where if bra-ket notation is used, this is

$$\backslash langle\; z\; \backslash mid\; A\; ~=~\; \backslash langle\; A\; z\; \backslash mid\; \backslash quad\; \backslash text\{\; for\; all\; \}\; z\; \backslash in\; H.$$

Normal operators

{{See also|Normal operator|Normal matrix}}

A continuous linear operator $A\; :\; H\; \backslash to\; H$ is called normal if $A\; A^*\; =\; A^*\; A,$ which happens if and only if for all $z,\; h\; \backslash in\; H,$

$$\backslash left\backslash langle\; A\; A^*\; z\backslash mid\; h\; \backslash right\backslash rangle\; =\; \backslash left\backslash langle\; A^*\; A\; z\backslash mid\; h\; \backslash right\backslash rangle.$$

Using ({{EquationNote|Adjoint-transpose}}) and unraveling notation and definitions produces the following characterization of normal operators in terms of inner products of continuous linear functionals: $A$ is a normal operator if and only if

{{NumBlk|:|$\backslash left\backslash langle\; \backslash ,\backslash langle\; A\; h\; \backslash mid\backslash cdot\backslash ,\; \backslash rangle\backslash mid\backslash langle\; A\; z\; \backslash mid\backslash cdot\backslash ,\; \backslash rangle\backslash ,\; \backslash right\backslash rangle\_\{H^*\}\; ~=~\; \backslash left\backslash langle\; \backslash ,\backslash langle\; h\; |\; A(\backslash cdot)\; \backslash rangle\backslash mid\backslash langle\; z\; \backslash mid\; A(\backslash cdot)\; \backslash rangle\backslash ,\; \backslash right\backslash rangle\_\{H^*\}\; \backslash quad\; \backslash text\{\; for\; all\; \}\; z,\; h\; \backslash in\; H$|{{EquationRef|Normality functionals}}|LnSty=1px dashed black}}

where the left hand side is also equal to $\backslash overline\{\backslash langle\; A\; h\; \backslash mid\; A\; z\; \backslash rangle\}\_H\; =\; \backslash langle\; A\; z\; \backslash mid\; A\; h\; \backslash rangle\_H.$

The left hand side of this characterization involves only linear functionals of the form $\backslash langle\; A\; h\; \backslash mid\backslash cdot\backslash ,\; \backslash rangle$ while the right hand side involves only linear functions of the form $\backslash langle\; h\; \backslash mid\; A(\backslash cdot)\; \backslash rangle$ (defined as above).

So in plain English, characterization ({{EquationNote|Normality functionals}}) says that an operator is normal when the inner product of any two linear functions of the first form is equal to the inner product of their second form (using the same vectors $z,\; h\; \backslash in\; H$ for both forms).

In other words, if it happens to be the case (and when $A$ is injective or self-adjoint, it is) that the assignment of linear functionals $\backslash langle\; A\; h\; \backslash mid\backslash cdot\backslash ,\; \backslash rangle\; ~\backslash mapsto~\; \backslash langle\; h\; |\; A(\backslash cdot)\; \backslash rangle$ is well-defined (or alternatively, if $\backslash langle\; h\; |\; A(\backslash cdot)\; \backslash rangle\; ~\backslash mapsto~\; \backslash langle\; A\; h\; \backslash mid\backslash cdot\backslash ,\; \backslash rangle$ is well-defined) where $h$ ranges over $H,$ then $A$ is a normal operator if and only if this assignment preserves the inner product on $H^*.$

The fact that every self-adjoint bounded linear operator is normal follows readily by direct substitution of $A^*\; =\; A$ into either side of $A^*\; A\; =\; A\; A^*.$

This same fact also follows immediately from the direct substitution of the equalities ({{EquationNote|Self-adjointness functionals}}) into either side of ({{EquationNote|Normality functionals}}).

Alternatively, for a complex Hilbert space, the continuous linear operator $A$ is a normal operator if and only if $\backslash |Az\backslash |\; =\; \backslash left\backslash |A^*\; z\backslash right\backslash |$ for every $z\; \backslash in\; H,${{sfn|Rudin|1991|pp=306-312}} which happens if and only if

$$\backslash |Az\backslash |\_H\; =\; \backslash |\backslash langle\; z\backslash ,\; |\; \backslash ,A(\backslash cdot)\; \backslash rangle\backslash |\_\{H^*\}\; \backslash quad\; \backslash text\{\; for\; every\; \}\; z\; \backslash in\; H.$$

Unitary operators

{{See also|Unitary transformation|Unitary matrix}}

An invertible bounded linear operator $A\; :\; H\; \backslash to\; H$ is said to be unitary if its inverse is its adjoint: $A^\{-1\}\; =\; A^*.$

By using ({{EquationNote|Adjoint-transpose}}), this is seen to be equivalent to $\backslash Phi\; \backslash circ\; A^\{-1\}\; =\; \{\}^\{t\}A\; \backslash circ\; \backslash Phi.$

Unraveling notation and definitions, it follows that $A$ is unitary if and only if

$$\backslash langle\; A^\{-1\}\; z\backslash mid\backslash cdot\backslash ,\; \backslash rangle\; =\; \backslash langle\; z\backslash mid\; A\; (\backslash cdot)\; \backslash rangle\; \backslash quad\; \backslash text\{\; for\; all\; \}\; z\; \backslash in\; H.$$

The fact that a bounded invertible linear operator $A\; :\; H\; \backslash to\; H$ is unitary if and only if $A^*\; A\; =\; \backslash operatorname\{Id\}\_H$ (or equivalently, $\{\}^t\; A\; \backslash circ\; \backslash Phi\; \backslash circ\; A\; =\; \backslash Phi$) produces another (well-known) characterization: an invertible bounded linear map $A$ is unitary if and only if

$$\backslash langle\; A\; z\backslash mid\; A\; (\backslash cdot)\backslash ,\; \backslash rangle\; =\; \backslash langle\; z\backslash mid\backslash cdot\backslash ,\; \backslash rangle\; \backslash quad\; \backslash text\{\; for\; all\; \}\; z\; \backslash in\; H.$$

Because $A\; :\; H\; \backslash to\; H$ is invertible (and so in particular a bijection), this is also true of the transpose $\{\}^t\; A\; :\; H^*\; \backslash to\; H^*.$ This fact also allows the vector $z\; \backslash in\; H$ in the above characterizations to be replaced with $A\; z$ or $A^\{-1\}\; z,$ thereby producing many more equalities. Similarly, $\backslash ,\backslash cdot\backslash ,$ can be replaced with $A(\backslash cdot)$ or $A^\{-1\}(\backslash cdot).$

• {{annotated link|Choquet theory}}
• {{annotated link|Covariance operator}}
• {{annotated link|Fundamental theorem of Hilbert spaces}}
• {{annotated link|Matrix coefficient}}

{{reflist}}

{{reflist|group=note|refs=

This means that for all vectors $y\; \backslash in\; H:$ (1) $\backslash Phi\; :\; H\; \backslash to\; H^*$ is injective. (2) The norms of $y$ and $\backslash Phi(y)$ are the same: $\backslash |\backslash Phi(y)\backslash |\; =\; \backslash |y\backslash |.$ (3) $\backslash Phi$ is an additive map, meaning that $\backslash Phi(x\; +\; y)\; =\; \backslash Phi(x)\; +\; \backslash Phi(y)$ for all $x,\; y\; \backslash in\; H.$ (4) $\backslash Phi$ is conjugate homogeneous: $\backslash Phi(s\; y)\; =\; \backslash overline\{s\}\; \backslash Phi(y)$ for all scalars $s.$ (5) $\backslash Phi$ is real homogeneous: $\backslash Phi(r\; y)\; =\; r\; \backslash Phi(y)$ for all real numbers $r\; \backslash in\; \backslash R.$

This footnote explains how to define - using only $H$'s operations - addition and scalar multiplication of affine hyperplanes so that these operations correspond to addition and scalar multiplication of linear functionals. Let $H$ be any vector space and let $H^\{\backslash \#\}$ denote its algebraic dual space. Let $\backslash mathcal\{A\}\; :=\; \backslash left\backslash \{\; \backslash varphi^\{-1\}(1)\; :\; \backslash varphi\; \backslash in\; H^\{\backslash \#\}\; \backslash right\backslash \}$ and let $\backslash ,\backslash hat\{\backslash cdot\}\backslash ,$ and $\backslash ,\backslash hat\{+\}\backslash ,$ denote the (unique) vector space operations on $\backslash mathcal\{A\}$ that make the bijection $I\; :\; H^\{\backslash \#\}\; \backslash to\; \backslash mathcal\{A\}$ defined by $\backslash varphi\; \backslash mapsto\; \backslash varphi^\{-1\}(1)$ into a vector space isomorphism. Note that $\backslash varphi^\{-1\}(1)\; =\; \backslash varnothing$ if and only if $\backslash varphi\; =\; 0,$ so $\backslash varnothing$ is the additive identity of $\backslash left(\backslash mathcal\{A\},\; \backslash hat\{+\},\; \backslash hat\{\backslash cdot\}\backslash right)$ (because this is true of $I^\{-1\}(\backslash varnothing)\; =\; 0$ in $H^\{\backslash \#\}$ and $I$ is a vector space isomorphism). For every $A\; \backslash in\; \backslash mathcal\{A\},$ let $\backslash ker\; A\; =\; H$ if $A\; =\; \backslash varnothing$ and let $\backslash ker\; A\; =\; A\; -\; A$ otherwise; if $A\; =\; I(\backslash varphi)\; =\; \backslash varphi^\{-1\}(1)$ then $\backslash ker\; A\; =\; \backslash ker\; \backslash varphi$ so this definition is consistent with the usual definition of the kernel of a linear functional. Say that $A,\; B\; \backslash in\; \backslash mathcal\{A\}$ are {{em|parallel}} if $\backslash ker\; A\; =\; \backslash ker\; B,$ where if $A$ and $B$ are not empty then this happens if and only if the linear functionals $I^\{-1\}(A)$ and $I^\{-1\}(B)$ are non-zero scalar multiples of each other. The vector space operations on the vector space of affine hyperplanes $\backslash mathcal\{A\}$ are now described in a way that involves {{em|only}} the vector space operations on $H$; this results in an interpretation of the vector space operations on the algebraic dual space $H^\{\backslash \#\}$ that is entirely in terms of affine hyperplanes. Fix hyperplanes $A,\; B\; \backslash in\; \backslash mathcal\{A\}.$ If $s$ is a scalar then $s\; \backslash hat\{\backslash cdot\}\; A\; =\; \backslash left\backslash \{\; h\; \backslash in\; H\; :\; s\; h\; \backslash in\; A\; \backslash right\backslash \}.$ Describing the operation $A\; \backslash hat\{+\}\; B$ in terms of only the sets $A\; =\; \backslash varphi^\{-1\}(1)$ and $B\; =\; \backslash psi^\{-1\}(1)$ is more complicated because by definition, $A\; \backslash hat\{+\}\; B\; =\; I(\backslash varphi)\; \backslash hat\{+\}\; I(\backslash psi)\; :=\; I(\backslash varphi\; +\; \backslash psi)\; =\; (\backslash varphi\; +\; \backslash psi)^\{-1\}(1).$ If $A\; =\; \backslash varnothing$ (respectively, if $B\; =\; \backslash varnothing$) then $A\; \backslash hat\{+\}\; B$ is equal to $B$ (resp. is equal to $A$) so assume $A\; \backslash neq\; \backslash varnothing$ and $B\; \backslash neq\; \backslash varnothing.$ The hyperplanes $A$ and $B$ are parallel if and only if there exists some scalar $r$ (necessarily non-0) such that $A\; =\; r\; B,$ in which case $A\; \backslash hat\{+\}\; B\; =\; \backslash left\backslash \{\; h\; \backslash in\; H\; :\; (1\; +\; r)\; h\; \backslash in\; B\; \backslash right\backslash \};$ this can optionally be subdivided into two cases: if $r\; =\; -1$ (which happens if and only if the linear functionals $I^\{-1\}(A)$ and $I^\{-1\}(B)$ are negatives of each) then $A\; \backslash hat\{+\}\; B\; =\; \backslash varnothing$ while if $r\; \backslash neq\; -1$ then $A\; \backslash hat\{+\}\; B\; =\; \backslash frac\{1\}\{1+r\}\; B\; =\; \backslash frac\{r\}\{1+r\}\; A.$ Finally, assume now that $\backslash ker\; A\; \backslash neq\; \backslash ker\; B.$ Then $A\; \backslash hat\{+\}\; B$ is the unique affine hyperplane containing both $A\; \backslash cap\; \backslash ker\; B$ and $B\; \backslash cap\; \backslash ker\; A$ as subsets; explicitly, $\backslash ker\; \backslash left(A\; \backslash hat\{+\}\; B\backslash right)\; =\; \backslash operatorname\{span\}\backslash left(A\; \backslash cap\; \backslash ker\; B\; -\; B\; \backslash cap\; \backslash ker\; A\backslash right)$ and $A\; \backslash hat\{+\}\; B\; =\; A\; \backslash cap\; \backslash ker\; B\; +\; \backslash ker\; \backslash left(A\; \backslash hat\{+\}\; B\backslash right)\; =\; B\; \backslash cap\; \backslash ker\; A\; +\; \backslash ker\; \backslash left(A\; \backslash hat\{+\}\; B\backslash right).$ To see why this formula for $A\; \backslash hat\{+\}\; B$ should hold, consider $H\; :=\; \backslash R^3,$ $A\; :=\; \backslash varphi^\{-1\}(1),$ and $B\; :=\; \backslash psi^\{-1\}(1),$ where $\backslash varphi(x,\; y,\; z)\; :=\; x$ and $\backslash psi(x,\; y,\; z)\; :=\; x\; +\; y$ (or alternatively, $\backslash psi(x,\; y,\; z)\; :=\; y$). Then by definition, $A\; \backslash hat\{+\}\; B\; :=\; (\backslash varphi\; +\; \backslash psi)^\{-1\}(1)$ and $\backslash ker\; \backslash left(A\; \backslash hat\{+\}\; B\backslash right)\; :=\; (\backslash varphi\; +\; \backslash psi)^\{-1\}(0).$ Now $A\; \backslash cap\; \backslash ker\; B\; ~=~\; \backslash varphi^\{-1\}(1)\; \backslash cap\; \backslash psi^\{-1\}(0)\; ~\backslash subseteq~\; (\backslash varphi\; +\; \backslash psi)^\{-1\}(1)$ is an affine subspace of codimension $2$ in $H$ (it is equal to a translation of the $z$-axis $\backslash \{(0,\; 0)\backslash \}\; \backslash times\; \backslash R$). The same is true of $B\; \backslash cap\; \backslash ker\; A.$ Plotting an $x$-$y$-plane cross section (that is, setting $z\; =$ constant) of the sets $\backslash ker\; A,\; \backslash ker\; B,\; A$ and $B$ (each of which will be plotted as a line), the set $(\backslash varphi\; +\; \backslash psi)^\{-1\}(1)$ will then be plotted as the (unique) line passing through the $A\; \backslash cap\; \backslash ker\; B$ and $B\; \backslash cap\; \backslash ker\; A$ (which will be plotted as two distinct points) while $(\backslash varphi\; +\; \backslash psi)^\{-1\}(0)\; =\; \backslash ker\; \backslash left(A\; \backslash hat\{+\}\; B\backslash right)$ will be plotted the line through the origin that is parallel to $A\; \backslash hat\{+\}\; B\; =\; (\backslash varphi\; +\; \backslash psi)^\{-1\}(1).$ The above formulas for $\backslash ker\; \backslash left(A\; \backslash hat\{+\}\; B\backslash right)\; :=\; (\backslash varphi\; +\; \backslash psi)^\{-1\}(0)$ and $A\; \backslash hat\{+\}\; B\; :=\; (\backslash varphi\; +\; \backslash psi)^\{-1\}(1)$ follow naturally from the plot and they also hold in general.

If $\backslash mathbb\{F\}\; =\; \backslash R$ then the inner product will be symmetric so it does not matter which coordinate of the inner product the element $y$ is placed into because the same map will result.

But if $\backslash mathbb\{F\}\; =\; \backslash Complex$ then except for the constant $0$ map, antilinear functionals on $H$ are completely distinct from linear functionals on $H,$ which makes the coordinate that $y$ is placed into is {{em|very}} important.

For a non-zero $y\; \backslash in\; H$ to induce a {{em|linear}} functional (rather than an {{em|anti}}linear functional), $y$ {{em|must}} be placed into the {{em|anti}}linear coordinate of the inner product. If it is incorrectly placed into the linear coordinate instead of the antilinear coordinate then the resulting map will be the antilinear map $h\; \backslash mapsto\; \backslash langle\; y\; ,\; h\; \backslash rangle\; =\; \backslash langle\; h\; \backslash mid\; y\; \backslash rangle,$ which is {{em|not}} a linear functional on $H$ and so it will {{em|not}} be an element of the continuous dual space $H^*.$

The notation $\backslash left\backslash langle\; z\; \backslash mid\; A(\backslash cdot)\; \backslash right\backslash rangle\_Z$ denotes the continuous linear functional defined by $g\; \backslash mapsto\; \backslash left\backslash langle\; z\; \backslash mid\; A\; g\; \backslash right\backslash rangle\_Z.$

}}

Proofs

{{reflist|group=proof|refs=

This is because $x\_K\; =\; x\; -\; \backslash frac\{\backslash left\backslash langle\; x,\; f\_\backslash varphi\; \backslash right\backslash rangle\}\{\backslash left\backslash |f\_\backslash varphi\backslash right\backslash |^2\}\; f\_\{\backslash varphi\}.$ Now use $\backslash left\backslash |\; f\_\{\backslash varphi\}\; \backslash right\backslash |^2\; =\; \backslash |\backslash varphi\backslash |^2$ and $\backslash left\backslash langle\; x,\; f\_\{\backslash varphi\}\; \backslash right\backslash rangle\; =\; \backslash varphi(x)$ and solve for $f\_\{\backslash varphi\}.$ $\backslash blacksquare$

$\backslash left\backslash langle\; A^*\; A\; z\; \backslash mid\; h\; \backslash right\backslash rangle\; =\; \backslash left\backslash langle\; \backslash ,A\; z\backslash mid\; A\; h\backslash ,\; \backslash right\backslash rangle\_H\; =\; \backslash left\backslash langle\; \backslash ,\backslash Phi\; A\; h\backslash mid\backslash Phi\; A\; z\backslash ,\; \backslash right\backslash rangle\_\{H^*\}$ where $\backslash Phi\; A\; h\; :=\; \backslash left\backslash langle\; A\; h\backslash mid\backslash cdot\backslash ,\; \backslash right\backslash rangle$ and $\backslash Phi\; A\; z\; :=\; \backslash left\backslash langle\; A\; z\; \backslash mid\backslash cdot\backslash ,\; \backslash right\backslash rangle.$

By definition of the adjoint, $\backslash left\backslash langle\; A^*\; h\backslash mid\; A^*\; z\backslash ,\; \backslash right\backslash rangle\; =\; \backslash left\backslash langle\; h\; \backslash mid\; A\; A^*\; z\backslash ,\; \backslash right\backslash rangle$ so taking the complex conjugate of both sides proves that $\backslash left\backslash langle\; A\; A^*\; z\backslash mid\; h\; \backslash right\backslash rangle\; =\; \backslash left\backslash langle\; A^*\; z\backslash mid\; A^*\; h\; \backslash right\backslash rangle.$

From $A^*\; =\; \backslash Phi^\{-1\}\; \backslash circ\; \{\}^\{t\}A\; \backslash circ\; \backslash Phi,$ it follows that $\backslash left\backslash langle\; A\; A^*\; z\backslash ,\; |\; \backslash ,h\; \backslash right\backslash rangle\_H$

= \left\langle A^* z\mid A^* h \right\rangle_H

= \left\langle \Phi^{-1} \circ {}^{t}A \circ \Phi z\mid\Phi^{-1} \circ {}^t A \circ \Phi h \right\rangle_H

= \left\langle \,{}^{t}A \circ \Phi h\mid{}^t A \circ \Phi z \right\rangle_{H^*} where $\backslash left(\{\}^\{t\}A\; \backslash circ\; \backslash Phi\backslash right)\; h\; =\; \backslash langle\; h\backslash ,\; |\; \backslash ,A\; (\backslash cdot)\; \backslash rangle$ and $\backslash left(\{\}^\{t\}A\; \backslash circ\; \backslash Phi\backslash right)\; z\; =\; \backslash langle\; z\backslash ,\; |\; \backslash ,A\; (\backslash cdot)\; \backslash rangle.$ $\backslash blacksquare$

}}

• {{Bachman Narici Functional Analysis 2nd Edition}}
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• {{citation | last=Roman | first=Stephen

| title=Advanced Linear Algebra | edition=Third | series=Graduate Texts in Mathematics | publisher = Springer | date=2008| pages= | isbn=978-0-387-72828-5 |author-link=Steven Roman}}

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• {{Trèves François Topological vector spaces, distributions and kernels}}

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