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 φx defined by
- φx(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