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(x_{1} + x_{2}) = Φ(x_{1}) + Φ(x_{2})
- 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.
All Wikipedia text
is available under the
terms of the GNU Free Documentation License