Riesz representation theorem

The Riesz representation theorem (after Frigyes Riesz ) is in mathematics, a statement of functional analysis, which characterizes the dual space of certain Banach spaces. Since Riesz was involved in several such sets, different sets are called Rieszscher representation theorem. Mostly, however, is meant the set of Riesz - Markov.

Motivation

In the functional analysis one gets information about the structure of Banach spaces from the study of linear, continuous functionals. For example, allowed the separation theorem, with their help to separate convex sets from each other. This therefore results in a natural task, the space of all such functionals - to study better - the dual space.

Dual spaces of normed vector spaces - and thus of Banach spaces - are always self- Banach spaces and by the theorem of Hahn- Banach not trivial: The functional is obviously always continuous and the theorem ensures the existence of further continuous functionals to. It is now an obvious idea to look for (isometric ) isomorphisms between a given dual space and a known, tangible space.

In finite-dimensional vector spaces, it is usually easy to characterize dual spaces: Consider the example of a functional from the dual space of the one referred to as. According to results of linear algebra can be represented as multiplication by a matrix

And consequently using the ( standard ) inner product as

One sees now easy one: The picture

Is bijective and isometric. Using the we can identify with the self that is the dual space.

The set of Fréchet - Riesz generalize this finding to general Hilbert spaces, while the Riesz - Markov the dual space of the space of continuous functions on a compact metric space characterized. Another known, associated with the name Riesz duality relationship is to identify the dual space of spaces with the spaces, wherein, see duality of rooms.

The set of Fréchet - Riesz

Statement

Be a Hilbert space. Then there exists for every continuous functional exactly, so that:

Conversely, for a given imaging

A continuous functional with operator norm.

Evidence

Existence: Let be a continuous linear functional.

If so choose.

Is, then its core is a closed subspace of. Therefore, and there is.

Now with his. Then is. Substituting, then.

After the homomorphism induces an isomorphism. Thus, in particular, each of the mold. is therefore

For the uniqueness, assume that there is another vector. Then, for every that. So in particular it follows for that.

The Riesz - Markov

The Riesz - Markov characterizes the dual space of continuous functions on a compact Hausdorff space. It represents the continuous linear functionals as integral dar. Specifically, he states:

Is a compact Hausdorff space and a continuous linear functional. Then a clear regular complex Borel measure exists, so that

Is satisfied for all. Conversely, by

For a given regular complex Borel measure defined on a continuous functional.

The proof even shows where the ( Banach ) space of regular complex Borel measures on with variation norm.

Duality of Lp- spaces

The set of Fréchet - Riesz, since every Hilbert space is isomorphic to a room, be regarded as a set - over spaces. It can be generalized to spaces. This denominated in brief phrase is often cited as a Riesz, rarer than Rieszscher representation theorem.