Embedding set by Arens-Eells

The Embedding Theorem of Arens-Eells ( English Arens-Eells embedding theorem ) is a mathematical theorem which is to be classified in the transition field between the mathematical sub-areas of analysis , functional analysis and topology . It goes back to the two mathematicians Richard Friederich Arens and James Eells and deals with the question of the embedding of any metric spaces in complex standardized spaces and especially in complex Banach spaces .

Formulation of the sentence

The sentence can be formulated as follows:

Be a metric space with a metric ${\displaystyle X}$

${\displaystyle d\colon X\times X\to \mathbb {R} ^{\geq 0}}$   .

Then:

${\displaystyle X}$ is isometrically embeddable into a normalized - vector space , the resultant under this isometric embedding image space of in the full vector space with respect standard topology a closed topological subspace is. ${\displaystyle \mathbb {C} }$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {V}}}$

Evidence and construction sketch

According to Väth's presentation, the proof can be given as follows:

The construction of the isometric embedding begins with the first isometric creation of a (not necessarily closed) subspace of a complex Banach space . In this the normalized vector space is then to be constructed as a - linear envelope of the image space defined. From this it is finally shown that it is closed with regard to the standard topology inherited from . ${\displaystyle \phi }$ ${\displaystyle X}$ ${\displaystyle {\mathcal {B}}}$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle {\langle \phi (X)\rangle }_{\mathbb {C} }}$ ${\displaystyle \phi (X)}$ ${\displaystyle {\mathcal {B}}}$

The construction of begins with the set system of all non-empty finite subsets of . ${\displaystyle {\mathcal {B}}}$ ${\displaystyle {\mathcal {E}}\subseteq {\mathcal {P}}{(X)}}$ ${\displaystyle X}$

Then you bet

${\displaystyle {\mathcal {B}}:=B({\mathcal {E}},\mathbb {C} )}$

as the function space of all bounded complex-valued functions . ${\displaystyle f\colon \,{\mathcal {E}}\to \mathbb {C} \;,\;E\mapsto z=f(E)\in \mathbb {C} }$

${\displaystyle {\mathcal {B}}}$is provided with the supremum norm

${\displaystyle f\mapsto \|f\|_{\infty }=\sup _{E\in {\mathcal {E}}}|f(E)|}$ ,

where in the body as always the complex amount ${\displaystyle \mathbb {C} }$

${\displaystyle |z|={\sqrt {z\cdot {\bar {z}}}}}$

is taken as a basis.

In a now element fixed. ${\displaystyle X}$ ${\displaystyle x_{0}\in X}$

With this one defines a mapping with the aid of the distance function belonging to the given metric ${\displaystyle d}$ ${\displaystyle D}$

${\displaystyle \phi \colon \,X\to {\mathcal {B}}\;,\;x\mapsto f_{x}\in {\mathcal {B}}}$  ,

by making the settlement

${\displaystyle f_{x}(E):=D(x,E)-D(x_{0},E)}$

makes, for because of the finiteness of always ${\displaystyle E\in {\mathcal {E}}\;,x\in X}$ ${\displaystyle E}$

${\displaystyle D(x,E)=\min _{e\in E}d(x,e)}$

applies.

It must be taken into account here that the distance function is lipschitz continuous with Lipschitz constant, i.e. always ${\displaystyle L=1}$

${\displaystyle |f_{x}(E)|\leq d(x,x_{0})}$

and thus every limited function. ${\displaystyle f_{x}}$

The image obtained in this way then turns out to be an isometry between and the image space with the desired properties. ${\displaystyle \phi }$ ${\displaystyle X}$ ${\displaystyle \phi (X)}$

Corollary

As a direct consequence of the derivation of the theorem it follows that every metric space has a metric completion . This can be constructed as a closed envelope within . ${\displaystyle X}$ ${\displaystyle {\hat {X}}}$ ${\displaystyle {\hat {X}}:={\overline {X}}^{\mathcal {B}}}$ ${\displaystyle {\mathcal {B}}}$

annotation

• Directly related to the Arens-Eells Theorem is the Kunugui Theorem , which was published by Kinjirô Kunugui in 1935 and which is based on the same ideas, but is somewhat weaker. Similar or related sentences were provided by Kazimierz Kuratowski (also in 1935), Menahem Wojdysławski (in 1939) and Victor Klee (in 1951).
• In the event that the metric is complete , the closure of the image space results directly from the completeness. ${\displaystyle X}$ ${\displaystyle d}$ ${\displaystyle \phi (X)}$
• In their publication of 1956, Arens and Eells showed, in addition to the theorem formulated above, but with the help of a similar proof approach, that even every uniform space , the structure of which is determined by pseudometrics with certain separation properties, can be completely embedded in a Hausdorffian topological vector space .

literature

• Richard F. Arens, James Eells, Jr .: On embedding uniform and topological spaces . In: Pacific J. Math . tape 6 , 1956, pp. 397-403 ( online , MR0081458 ).
• VL Klee, Jr .: Some characterizations of compactness . In: The American Mathematical Monthly . tape 58 , 1951, pp. 389-393 , JSTOR : 2306551 ( MR0042682 ).
• Casimir Kuratowski: Quelquesproblemèmes concernant les espaces métriques non-séparables . In: Fund. Math . tape 25 , 1935, pp. 534-545 .
• M. Wojdysławski: Rétractes absolus et hyperespaces des continus . In: Fund. Math . tape 32 , 1939, pp. 184-192 .
• Horst Schubert : Topology . 4th edition. BG Teubner Verlag, Stuttgart 1975, ISBN 3-519-12200-6 ( MR0423277 ).
• Martin Väth : Topological Analysis . From the Basics to the Triple Degree for Nonlinear Fredholm Inclusions (=  De Gruyter Series in Nonlinear Analysis and Applications . Volume 16 ). Verlag Walter de Gruyter, Berlin / Boston 2012, ISBN 978-3-11-027722-7 ( MR2961860 ).
• James H. Wells, Lynn R. Williams: Embeddings and Extensions in Analysis (=  results of mathematics and their border areas . Volume 84 ). Springer Verlag, Berlin (among others) 1975, ISBN 3-540-07067-2 ( MR0461107 ).

References and comments

1. ↑ ^{a b c} Väth: Topological Analysis. 2012, p. 89 ff.
2. ^ Wells, Williams: Embeddings and Extensions in Analysis. 1975, p. 1.
3. ↑ Horst Schubert : Topology . 4th edition. BG Teubner Verlag, Stuttgart 1975, ISBN 3-519-12200-6 , pp. 78 ( MR0423277 ).
4. ^ Väth: Topological Analysis. 2012, p. 91.