# 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. ${\ displaystyle x \ mapsto 0}$

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}$${\ displaystyle \ langle v, ww ^ {\ prime} \ rangle = 0}$${\ displaystyle v = ww ^ {\ prime}}$${\ displaystyle 0 = \ langle ww ^ {\ prime}, ww ^ {\ prime} \ rangle = \ | ww ^ {\ prime} \ | ^ {2}}$${\ displaystyle w = w ^ {\ prime}}$

## 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}}$

## 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).