Let H be a Hilbert space, and let H* denote its dual space, consisting of all continuous linear functionals from H into the field or . If is an element of H, then the function for all in H defined by:
where 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.
Riesz–Fréchet representation theorem. Let be a Hilbert space and . Then there exists such that for any , . Moreover
Proof. Let . Clearly is closed subspace of . If , then we can trivially choose . Now assume . Then is one-dimensional. Indeed, let be nonzero vectors in . Then there is nonzero real number , such that . Observe that and , so . This means that . Now let be unit vector in . For arbitrary , let be the orthogonal projections of onto respectively. Then and (from the properties of orthogonal projections), so that and . Thus . Hence . We also see . From the Cauchy-Bunyakovsky-Schwartz inequality, thus for with unit norm . This implies that .
Given any continuous linear functional g in H*, the corresponding element can be constructed uniquely by , where is an orthonormal basis of H, and the value of does not vary by choice of basis. Thus, if , then
Theorem. The mapping : H → H* defined by = is an isometric (anti-) isomorphism, meaning that:
If the base field is , then for all real numbers λ.
If the base field is , then for all complex numbers λ, where denotes the complex conjugation of .
The inverse map of can be described as follows. Given a non-zero element of H*, the orthogonal complement of the kernel of is a one-dimensional subspace of H. Take a non-zero element z in that subspace, and set . Then = .
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. The theorem says that, every bra has a corresponding ket , and the latter is unique.