Riesz representation theorem

For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.

In functional analysis the Riesz representation theorem describes the dual of a Hilbert space. It is named in honour of Frigyes Riesz.

Statement

If $H$ is a Hilbert Space, then the dual space of $H$, denoted by $H^{*}$, is given by

$H^{*}=B(H,\mathbb {F} )=\{\{({\underline {y}},\langle {\underline {y}},{\underline {x}}\rangle ):{\underline {y}}\in H\}:{\underline {x}}\in H\}$.

Here, $B(H,\mathbb {F} )$ denotes the set of all bounded linear operators from $H$ to the set of scalars, $\mathbb {F}$, either real or complex, and $\langle \cdot ,\cdot \rangle :H\times H:\to \mathbb {F} ,$ denotes the endowed inner product.

Some remarks

Note that for any inner product space, the mapping,

$T:H\to H^{*},$,

defined (as a function) by

$T({\underline {x}})({\underline {y}}):=\langle {\underline {y}},{\underline {x}}\rangle \in \mathbb {F} ,$

defines an element in the dual space. The statement of this theorem is that every element (function) in the dual can be cast as an inner product against some element in the original space.

Proof

"$\supseteq$": linearity comes from the fact that the inner product, by definition, is linear in the first argument and boundedness comes from the Cauchy-Schwartz inequality.

Now to deal with "$\subseteq$": Let ${\underline {\phi }}\in H'$; if ${\underline {\phi }}={\underline {0}}_{H'}$ then ${\underline {\phi }}=\{({\underline {y}},\langle {\underline {y}},{\underline {0}}_{H}\rangle ):{\underline {y}}\in H\}$.

Now suppose ${\underline {\phi }}\neq {\underline {0}}_{H'}$ and let ${\underline {x}}\in H$; ${\overline {ker({\underline {\phi }})}}=ker({\underline {\phi }})$ and $ker({\underline {\phi }})\leqslant H$ so by the Hilbert projection theorem $H=ker({\underline {\phi }})\oplus (ker({\underline {\phi }}))^{\perp }$.

Well, $ker(\phi )\neq H$ so $(ker({\underline {\phi }}))^{\perp }\neq \{{\underline {0}}_{H}\}$ so let ${\underline {z}}\in \{{\underline {z}}\in (ker({\underline {\phi }}))^{\perp }:\|{\underline {z}}\|_{H}=1\}$

Then by the linearity of ${\underline {\phi }}$, ${\underline {z}}{\underline {\phi }}({\underline {x}})-{\underline {x}}{\underline {\phi }}({\underline {z}})\in ker({\underline {\phi }})$, and so $\langle {\underline {z}}{\underline {\phi }}({\underline {x}})-{\underline {x}}{\underline {\phi }}({\underline {z}}),{\underline {z}}\rangle =0$ and so ${\underline {\phi }}({\underline {x}})\|{\underline {z}}\|_{H}=\langle {\underline {x}}{\underline {\phi }}({\underline {z}}),{\underline {z}}\rangle$ so ${\underline {\phi }}({\underline {x}})=\langle {\underline {x}},{\underline {z}}{\overline {{\underline {\phi }}({\underline {z}})}}\rangle$ as $\|{\underline {z}}\|_{H}=1$.

So ${\underline {\phi }}=\{({\underline {y}},\langle {\underline {y}},{\underline {z}}{\overline {{\underline {\phi }}({\underline {z}})}}\rangle ):{\underline {y}}\in H\}\in \{\{({\underline {y}},\langle {\underline {y}},{\underline {x}}\rangle ):{\underline {y}}\in H\}:{\underline {x}}\in H\}$

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 field is the complex numbers, the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural one as will be described next.

Let H be a Hilbert space, and let H* denote its dual space, consisting of all continuous linear functionals from H into the field R or C. If x is an element of H, then the function φ_x, defined by

$\phi _{x}(y)=\left\langle y,x\right\rangle \quad \forall y\in H$

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.

Theorem. The mapping Φ: H → H* defined by Φ(x) = φ_x is an isometric (anti-) isomorphism, meaning that:

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

The inverse map of Φ can be described as follows. Given an element φ of H*, the orthogonal complement of the kernel of φ 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 Φ(x) = φ.

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. When the theorem holds, every ket $|\psi \rangle$ has a corresponding bra $\langle \psi |$, and the correspondence is unambiguous.

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.
• J. D. Gray, The shaping of the Riesz representation theorem: A chapter in the history of analysis, Archive for History in the Exact Sciences, Vol 31(2) 1984–85, 127–187.
• 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).
• D. G. Hartig, The Riesz representation theorem revisited, American Mathematical Monthly, 90(4), 277–280 (A category theoretic presentation as natural transformation).
• "Riesz representation theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Walter Rudin, Real and Complex Analysis, McGraw-Hill, 1966, ISBN 0-07-100276-6.
• Weisstein, Eric W. "Riesz Representation Theorem". MathWorld.
• "Proof of Riesz representation theorem for separable Hilbert spaces". PlanetMath.
• Proof of Riesz representation theorem in Hilbert spaces on Bourbawiki