Representation set by Fréchet-Riesz

This article deals with the representation of the dual space of a Hilbert space. For the representation of positive linear forms on function spaces by means of dimensions, see Riesz-Markow's representation theorem .

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. ${\displaystyle x\mapsto 0}$

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: ${\displaystyle f}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle (\mathbb {R} ^{2})'}$

${\displaystyle x\mapsto {\begin{pmatrix}f_{1}&f_{2}\end{pmatrix}}x}$

and consequently also as a using the standard scalar product

${\displaystyle x\mapsto \langle {\vec {f}},x\rangle \;.}$

The image

${\displaystyle {\begin{aligned}\Phi \colon \mathbb {R} ^{2}&\to (\mathbb {R} ^{2})'\\{\vec {f}}&\mapsto \langle {\vec {f}},\cdot \rangle \end{aligned}}}$

is bijective and isometric . With the help of we can identify the dual space of the with the self. ${\displaystyle \Phi }$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \mathbb {R} ^{2}}$

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 . ${\displaystyle C^{0}(K)}$ ${\displaystyle K}$ ${\displaystyle L^{p}}$ ${\displaystyle L^{q}}$ ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$ ${\displaystyle L^{p}}$

statement

Be a Hilbert dream. Then there is exactly one for every continuous functional , so that: ${\displaystyle H}$ ${\displaystyle \alpha \in H'}$ ${\displaystyle w\in H}$

${\displaystyle {\begin{aligned}\alpha (v)&=\langle v,w\rangle ~~\forall \,v\in H\\\|\alpha \|&=\|w\|\end{aligned}}}$

The reverse is the case for the given figure ${\displaystyle w\in H}$

${\displaystyle v\mapsto \langle v,w\rangle }$

a continuous functional with operator norm . ${\displaystyle \|w\|}$

proof

Existence: Be a continuous, linear functional. ${\displaystyle \alpha \colon H\rightarrow \mathbb {C} }$

Is , so you choose . ${\displaystyle \alpha =0}$ ${\displaystyle w=0}$

Is , then its core is a closed subspace of . With the projection theorem it follows that . Since also follows . ${\displaystyle \alpha \neq 0}$ ${\displaystyle U:=\operatorname {Ker} (\alpha )}$ ${\displaystyle H}$ ${\displaystyle H=U\oplus U^{\perp }}$ ${\displaystyle U\neq H}$ ${\displaystyle U^{\perp }\neq \{0\}}$

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. ${\displaystyle w_{0}\in U^{\perp }}$ ${\displaystyle \|w_{0}\|=1}$ ${\displaystyle \alpha (w_{0})=c\neq 0}$ ${\displaystyle \lambda \in \mathbb {C} }$ ${\displaystyle \alpha }$ ${\displaystyle \alpha (\lambda w_{0})=\lambda c}$ ${\displaystyle \alpha }$ ${\displaystyle \operatorname {span} \{w_{0}\}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \alpha }$ ${\displaystyle U^{\perp }=H/\operatorname {Ker} (\alpha )}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle U^{\perp }=\operatorname {span} \{w_{0}\}}$ ${\displaystyle v\in H}$ ${\displaystyle v=\lambda w_{0}+u}$ ${\displaystyle u\in U}$ ${\displaystyle \lambda \in \mathbb {C} }$ ${\displaystyle \alpha (v)=\alpha (\lambda w_{0}+u)=\lambda c}$ ${\displaystyle w=cw_{0}}$ ${\displaystyle u\perp w}$ ${\displaystyle \langle v,w\rangle =\langle \lambda w_{0},w\rangle =\lambda c}$ ${\displaystyle \alpha (v)=\langle v,w\rangle }$

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 . ${\displaystyle w^{\prime }}$ ${\displaystyle \alpha (v)=\langle v,w^{\prime }\rangle }$ ${\displaystyle v\in H}$ ${\displaystyle \langle v,w-w^{\prime }\rangle =0}$ ${\displaystyle v=w-w^{\prime }}$ ${\displaystyle 0=\langle w-w^{\prime },w-w^{\prime }\rangle =\|w-w^{\prime }\|^{2}}$ ${\displaystyle w=w^{\prime }}$

Duality of L ^p spaces

→ Main article : Duality of L ^p spaces

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 . ${\displaystyle L^{2}}$ ${\displaystyle L^{2}}$ ${\displaystyle L^{p}}$ ${\displaystyle (L^{p})\,'\cong L^{q}}$

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

1. ↑ 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).