Riesz representation theorem


Riesz representation theorem, sometimes called Riesz–Fréchet representation theorem, named after Frigyes Riesz and Maurice René Fréchet, establishes an important connection between a Hilbert space and its continuous dual space. If the underlying field is the real numbers, the two are isometrically isomorphic; if the underlying field is the complex numbers, the two are isometrically anti-isomorphic. The isomorphism is a particular natural one as will be described next; a natural isomorphism.

Notation

Let H be a Hilbert space, and let H* denote its dual space, consisting of all bounded operators 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 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. We claim that is one-dimensional. To see this, let be nonzero vectors in. Then, and there must be a nonzero real number, such that. Observe that and, so. This means that. Now let be unit vector in. For arbitrary, let be the orthogonal projection of onto. Then and , 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

Properties

The mapping : H → H* defined by = is an isometric isomorphism, meaning that:
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.
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.