# Representation set by Fréchet-Riesz

The **presentation set of Fréchet Riesz** , sometimes **set of Fréchet Riesz** or **Riesz representation theorem** or **representation theorem of Riesz** (by Frigyes Riesz ) is in mathematics , a statement of functional analysis that the dual space of certain Banach spaces characterized. Since Riesz was involved in several such sentences, different sentences are called Riesz's representation sentence.

## motivation

Functional analysis provides information about the structure of Banach spaces from the study of linear , continuous functionals . For example, allows the separation rate , with their help convex sets under certain conditions to separate from each other. It is therefore a natural task to study the space of all such continuous functionals - the dual space - more closely.

Dual spaces of normalized vector spaces - and thus also of Banach spaces - are always Banach spaces themselves. The constant functional is evidently always continuous and Hahn-Banach's theorem ensures the existence of "many" other continuous functionals. However, this existence theorem is purely abstract and is based on non- constructive methods such as Zorn's Lemma . It is now obvious to search for isometric isomorphisms between a known space and the dual space to be examined in order to describe the latter in a tangible way.

In finite-dimensional vector spaces it is easy to characterize dual spaces: As an example, consider a functional from the dual space of , which is called. According to the results of linear algebra , it can be represented by multiplying by a row vector from the left:

and consequently also as a using the standard scalar product

The image

is bijective and isometric . With the help of we can identify the dual space of the with the self.

The *Fréchet-Riesz theorem* generalizes this knowledge to general Hilbert spaces , while Riesz-Markow's representation theorem characterizes the dual space of , the space of continuous functions on a compact Hausdorff space . Another well-known duality relationship associated with the name Riesz is the identification of the dual spaces of -spaces with the spaces , whereby , see duality of -spaces .

## statement

Be a Hilbert dream. Then there is exactly one for every continuous functional , so that:

The reverse is the case for the given figure

a continuous functional with operator norm .

### proof

Existence: Be a continuous, linear functional.

Is , so you choose .

Is , then its core is a closed subspace of . With the projection theorem it follows that . Since also follows .

Choose with . Then is . For it now follows from the linearity of that . In particular, represents an isomorphism between and . According to the homomorphism law, there is also an isomorphism between and . Because of this, it follows . Now each is of the form with and . Hence is . If one sets now , then applies and therefore . We conclude that it holds.

For the sake of clarity it is assumed that there is another vector with . Then applies to each that . If one sets , it follows in particular that .

##
Duality of *L *^{p} spaces

^{p}

Since every Hilbert space is isomorphic to a -space, Fréchet-Riesz's theorem can be viewed as a theorem about -spaces. It can be generalized to spaces . This sentence, in short form , is often quoted as a *sentence by Riesz* , more rarely as *Riesz's representation **sentence* .

## literature

- Dirk Werner : Functional Analysis . 6th corrected edition. Springer, Berlin 2007, ISBN 978-3-540-72533-6 .
- Hans Wilhelm Alt: Linear Functional Analysis . 5th edition. Springer, Berlin, Heidelberg, New York 2006, ISBN 978-3-540-34186-4 .
- Friedrich Sauvigny : Partial differential equations of geometry and physics. Basics and integral representations . tape 1 . Springer, 2004, ISBN 3-540-20453-9 .

## Individual evidence

- ↑ Dirk Werner : Functional Analysis . 6th corrected edition. Springer, Berlin 2007, ISBN 978-3-540-72533-6 , pp. 58 ff . (Corollary II.2.2 / 4/5).