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

φx(y)=⟨y,x⟩,{displaystyle varphi _{x}(y)=leftlangle y,xrightrangle ,}

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. Given any continuous linear functional g in H*, the corresponding element xg∈H{displaystyle x_{g}in H} can be constructed uniquely by xg=g(e1)e1+g(e2)e2+...{displaystyle x_{g}=g(e_{1})e_{1}+g(e_{2})e_{2}+...}, where {ei}{displaystyle {e_{i}}} is an orthonormal basis of H, and the value of xg{displaystyle x_{g}} does not vary by choice of basis. Thus, if y∈H,y=a1e1+a2e2+...{displaystyle yin H,y=a_{1}e_{1}+a_{2}e_{2}+...}, then g(y)=a1g(e1)+a2g(e2)+...=⟨xg,y⟩.{displaystyle g(y)=a_{1}g(e_{1})+a_{2}g(e_{2})+...=langle x_{g},yrangle .}

Theorem. The mapping Φ{displaystyle Phi }: H → H* defined by Φ(x){displaystyle Phi (x)} = φx{displaystyle varphi _{x}} is an isometric (anti-) isomorphism, meaning that:

• 
  Φ{displaystyle Phi } is bijective.


• The norms of x{displaystyle x} and φx{displaystyle varphi _{x}} agree: ‖x‖=‖Φ(x)‖{displaystyle Vert xVert =Vert Phi (x)Vert }.


• 
  Φ{displaystyle Phi } is additive: Φ(x1+x2)=Φ(x1)+Φ(x2){displaystyle Phi (x_{1}+x_{2})=Phi (x_{1})+Phi (x_{2})}.


• If the base field is R{displaystyle mathbb {R} }, then Φ(λx)=λΦ(x){displaystyle Phi (lambda x)=lambda Phi (x)} for all real numbers λ.


• If the base field is C{displaystyle mathbb {C} }, then Φ(λx)=λ¯Φ(x){displaystyle Phi (lambda x)={bar {lambda }}Phi (x)} for all complex numbers λ, where λ¯{displaystyle {bar {lambda }}} denotes the complex conjugation of λ{displaystyle lambda }.

The inverse map of Φ{displaystyle Phi } can be described as follows. Given a non-zero element φ{displaystyle varphi } of H*, the orthogonal complement of the kernel of φ{displaystyle varphi } is a one-dimensional subspace of H. Take a non-zero element z in that subspace, and set x=φ(z)¯⋅z/‖z‖2{displaystyle x={overline {varphi (z)}}cdot z/{leftVert zrightVert }^{2}}. Then Φ(x){displaystyle Phi (x)} = φ{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 ⟨ψ|{displaystyle langle psi |} has a corresponding ket |ψ⟩{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, .mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
  ISBN 0-07-100276-6.


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  "Proof of Riesz representation theorem for separable Hilbert spaces". PlanetMath.