# 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).