# Riesz representation theorem

There are several well-known theorems in functional analysis known as the Riesz representation theorem. They are named in honor of Frigyes Riesz.

This article will describe his theorem concerning the dual of a Hilbert space, which is sometimes called the Fréchet–Riesz theorem. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.

## The Hilbert space representation theorem

This theorem establishes an important connection between a Hilbert space and its continuous dual space. If the underlying field is the real numbers, the two are isometrically isomorphic; if the underlying field is the complex numbers, the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural one as will be described next; a natural isomorphism.

Let H be a Hilbert space, and let H* denote its dual space, consisting of all continuous linear functionals from H into the field $\mathbb {R}$  or $\mathbb {C}$ . If $x$  is an element of H, then the function $\varphi _{x},$  for all $y$  in H defined by:

$\varphi _{x}(y)=\left\langle y,x\right\rangle ,$

where $\langle \cdot ,\cdot \rangle$  denotes the inner product of the Hilbert space, is an element of H*. The Riesz representation theorem states that every element of H* can be written uniquely in this form.

Riesz-Fréchet representation theorem. Let $H$  be a Hilbert space and $\varphi \in H^{*}$ . Then there exists $f\in H$  such that for any $x\in H$ , $\varphi (x)=\langle f,x\rangle$ . Moreover $\|f\|_{H}=\|\varphi \|_{H*}$

Proof. Let $M=\{u\in H\ |\ \varphi (u)=0\}$ . Clearly $M$  is closed subspace of $H$ . If $M=H$ , then we can trivially choose $v=0$ . Now assume $M\neq H$ . Then $M^{\perp }$  is one-dimensional. Indeed, let $v_{1},v_{2}$  be nonzero vectors in $M^{\perp }$ . Then there is nonzero real number $\lambda$ , such that $\lambda \varphi (v_{1})=\varphi (v_{2})$ . Observe that $\lambda v_{1}-v_{2}\in M^{\perp }$  and $\varphi (\lambda v_{1}-v_{2})=0$ , so $\lambda v_{1}-v_{2}\in M$ . This means that $\lambda v_{1}-v_{2}=0$ . Now let $g$  be unit vector in $M^{\perp }$ . For arbitrary $x\in H$ , let $v$  be the orthogonal projections of $x$  onto $M^{\perp }$  respectively. Then $v=\langle g,v\rangle g$  and $\langle g,x-v\rangle =0$  (from the properties of orthogonal projections), so that $x-v\in M$  and $\langle g,x\rangle =\langle g,v\rangle$ . Thus $\varphi (x)=\varphi (v+x-v)=\varphi (\langle g,v\rangle g)+\varphi (x-v)=\langle g,v\rangle \varphi (g)=\langle g,x\rangle \varphi (g)$ . Hence $f=\varphi (g)g$ . We also see $\|f\|_{H}=\varphi (g)$ . From the Cauchy-Bunyakovsky-Schwartz inequality $\varphi (x)\leq \|g\|\|x\|\varphi (g)$ , thus for $x$  with unit norm $\varphi (x)\leq \varphi (g)$ . This implies that $\|\varphi \|_{H*}=\varphi (g)$ .

Given any continuous linear functional g in H*, the corresponding element $x_{g}\in H$  can be constructed uniquely by $x_{g}=g(e_{1})e_{1}+g(e_{2})e_{2}+...$ , where $\{e_{i}\}$  is an orthonormal basis of H, and the value of $x_{g}$  does not vary by choice of basis. Thus, if $y\in H,y=a_{1}e_{1}+a_{2}e_{2}+...$ , then $g(y)=a_{1}g(e_{1})+a_{2}g(e_{2})+...=\langle x_{g},y\rangle .$

Theorem. The mapping $\Phi$ : HH* defined by $\Phi (x)$  = $\varphi _{x}$  is an isometric (anti-) isomorphism, meaning that:

• $\Phi$  is bijective.
• The norms of $x$  and $\varphi _{x}$  agree: $\Vert x\Vert =\Vert \Phi (x)\Vert$ .
• $\Phi$  is additive: $\Phi (x_{1}+x_{2})=\Phi (x_{1})+\Phi (x_{2})$ .
• If the base field is $\mathbb {R}$ , then $\Phi (\lambda x)=\lambda \Phi (x)$  for all real numbers λ.
• If the base field is $\mathbb {C}$ , then $\Phi (\lambda x)={\bar {\lambda }}\Phi (x)$  for all complex numbers λ, where ${\bar {\lambda }}$  denotes the complex conjugation of $\lambda$ .

The inverse map of $\Phi$  can be described as follows. Given a non-zero element $\varphi$  of H*, the orthogonal complement of the kernel of $\varphi$  is a one-dimensional subspace of H. Take a non-zero element z in that subspace, and set $x={\overline {\varphi (z)}}\cdot z/{\left\Vert z\right\Vert }^{2}$ . Then $\Phi (x)$  = $\varphi$ .

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

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 $\langle \psi |$  has a corresponding ket $|\psi \rangle$ , and the latter is unique.