# 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 ${\displaystyle \mathbb {R} }$ or ${\displaystyle \mathbb {C} }$. If ${\displaystyle x}$ is an element of H, then the function ${\displaystyle \varphi _{x},}$ for all ${\displaystyle y}$ in H defined by:

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

where ${\displaystyle \langle \cdot ,\cdot \rangle }$ 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 ${\displaystyle H}$ be a Hilbert space and ${\displaystyle \varphi \in H^{*}}$. Then there exists ${\displaystyle f\in H}$ such that for any ${\displaystyle x\in H}$, ${\displaystyle \varphi (x)=\langle f,x\rangle }$. Moreover ${\displaystyle \|f\|_{H}=\|\varphi \|_{H*}}$

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

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

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

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

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

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

## References

• M. Fréchet (1907). Sur les ensembles de fonctions et les opérations linéaires. C. R. Acad. Sci. Paris 144, 1414–1416.
• F. Riesz (1907). Sur une espèce de géométrie analytique des systèmes de fonctions sommables. C. R. Acad. Sci. Paris 144, 1409–1411.
• F. Riesz (1909). Sur les opérations fonctionnelles linéaires. C. R. Acad. Sci. Paris 149, 974–977.
• P. Halmos Measure Theory, D. van Nostrand and Co., 1950.
• P. Halmos, A Hilbert Space Problem Book, Springer, New York 1982 (problem 3 contains version for vector spaces with coordinate systems).
• Walter Rudin, Real and Complex Analysis, McGraw-Hill, 1966, ISBN 0-07-100276-6.
• "Proof of Riesz representation theorem for separable Hilbert spaces". PlanetMath.