Table of contents

1 The Hilbert space representation theorem 2 The representation theorem for the dual of C0(X) 3 References

The Hilbert space representation theorem

This theorem 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

Φ : H -> H '

• 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 theorem was proven simultaneously by Riesz and Fréchet in 1907 (see references).

The representation theorem for the dual of C₀(X)

If X is a locally compact Hausdorff space, this theorem gives a concrete realisation of the dual space of , the set of continuous functions on X which vanish at infinity. It says that each linear functional in the dual space is given by integration against some bounded Lebesgue measure on X; so the dual space of can be identified with the space of such measures.

If a linear functional is positive, then the corresponding measure is also positive.

See also the entry on Mathworld.

References