Red–black tree

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

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

${\varphi }_x(y)=⟨y,x⟩\mathrm{\forall }y\in H$

where $⟨\cdot ,\cdot ⟩$ 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: $\Vert x\Vert =\Vert \mathrm{\Phi }(x)\Vert$.
• Φ is additive: $\mathrm{\Phi }(x_1+x_2)=\mathrm{\Phi }(x_1)+\mathrm{\Phi }(x_2)$.
• If the base field is R, then $\mathrm{\Phi }(\lambda x)=\lambda \mathrm{\Phi }(x)$ for all real numbers λ.
• If the base field is C, then $\mathrm{\Phi }(\lambda x)=\overline{\lambda }\mathrm{\Phi }(x)$ for all complex numbers λ, where $\overline{\lambda }$ 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 $x=\overline{\phi (z)}\cdot z/{\Vert z\Vert }^2$. 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 $|\psi ⟩$ has a corresponding bra $⟨\psi |$, 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).
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