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Riesz representation theorem

The Riesz representation theorem in functional analysis establishes an important connection between a Hilbert space and its dual space: if the ground field is the real numbers, the two are isometrically isomorphic; if the ground field is the complex numbers, the two are isometrically anti-isomorphic. The theorem is the justification for the bra-ket notation popular in the mathematical treatment of quantum mechanics. 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 functions from H into the base field R or C. If x is an element of H, then φ_x defined by

φ_x(y) = <x, y> for all y in H

is an element of H '. The Riesz representation theorem states that every element of H ' can be written in this form, and that furthermore the assignment Φ(x) = φ_x defines an isometric (anti-) isomorphism

Φ : H -> H '

meaning that

Φ is bijective
The norms of x and Φ(x) agree: ||x|| = ||Φ(x)||
Φ is additive: Φ(x₁ + x₂) = Φ(x₁) + Φ(x₂)
If the base field is R, then Φ(λ x) = λ Φ(x) for all real numbers λ
If the base field is C, then Φ(λ x) = λ^* Φ(x) for all complex numbers λ, where λ^* 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 = φ(z) / ||z||² · z. Then Φ(x) = φ.

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