The Riesz representation theorem, sometimes called the Riesz–Fréchet representation theorem after Frigyes Riesz and Maurice René Fréchet, establishes an important connection between a Hilbert space and its continuous dual space.
Real and complex Hilbert spaces have in common many, but by no means all, properties and results/theorems.
This article is intended for both mathematicians and physicists and will describe the theorem for both.
), then there is a crucial difference between the notations prevailing in mathematics versus physics, regarding which of the two variables is linear.
), the inner product is a symmetric map that is linear in each coordinate (bilinear), so there can be no such confusion.
satisfies the parallelogram law, which means that the polarization identity can be used to define a canonical inner product on
) and the usual dual norm (defined as the supremum over the closed unit ball).
The Hilbert projection theorem guarantees that for any nonempty closed convex subset
[note 1] Furthermore, the length of the representation vector is equal to the norm of the functional:
The Riesz representation theorem states that this map is surjective (and thus bijective) when
can be interpreted as being the affine hyperplane[note 3] that is parallel to the vector subspace
In the mathematical treatment of quantum mechanics, the theorem can be seen as a justification for the popular bra–ket notation.
Historically, the theorem is often attributed simultaneously to Riesz and Fréchet in 1907 (see references).
defined in the corollary to the Riesz representation theorem is the assignment that sends
(see the article about the polarization identity for additional details about this relationship).
The formula expressing a linear functional in terms of its real part is
on a complex Hilbert space is equal to the Riesz representation of its real part
is the equal to the vector obtained by using the origin complex Hilbert space
and so by the Riesz representation theorem, there exists a unique vector in
is necessarily a continuous (equivalently, a bounded) linear operator.
Alternatively, the value of the left and right hand sides of (Adjoint-transpose) at any given
satisfies certain defining conditions related to its adjoint, which was shown by (Adjoint-transpose) to essentially be just the transpose
Unraveling notation and definitions produces the following characterization of self-adjoint operators in terms of the aforementioned continuous linear functionals:
Using (Adjoint-transpose) and unraveling notation and definitions produces[proof 2] the following characterization of normal operators in terms of inner products of continuous linear functionals:
The left hand side of this characterization involves only linear functionals of the form
while the right hand side involves only linear functions of the form
So in plain English, characterization (Normality functionals) says that an operator is normal when the inner product of any two linear functions of the first form is equal to the inner product of their second form (using the same vectors
The fact that every self-adjoint bounded linear operator is normal follows readily by direct substitution of
Alternatively, for a complex Hilbert space, the continuous linear operator
) produces another (well-known) characterization: an invertible bounded linear map