Eberlein-compact room

Eberlein-compact rooms , named after William Frederick Eberlein , are examined in the mathematical sub-area of functional analysis. These are those compact spaces that appear as weakly compact subsets of a Banach space .

definition

A compact Hausdorff space is called Eberlein compact if it is homeomorphic to a weakly compact subset of a Banach space in the relatively weak topology.

A compact Hausdorff space is uniformly called Eberlein-compact if it is homeomorphic to a weakly compact subset of a Hilbert space in the relatively weak topology.

Since Hilbert spaces are special Banach spaces, the uniform Eberlein compactness is a stronger property than the Eberlein compactness.

Examples

• The unit sphere of a reflexive Banach space is Eberlein-compact, because the weak compactness of the unit sphere is one of the equivalent characterizations of reflexivity.
• Norm- compact subsets of a Banach space are Eberlein-compact, because such sets are also weakly compact.
• The Hilbert cube is uniformly Eberlein-compact because it is homeomorphic to a weakly compact subset of the Hilbert space . ${\displaystyle [0,1]^{\infty }}$ ${\displaystyle \ell ^{2}}$

${\displaystyle \varphi :[0,1]^{\infty }\rightarrow \{(\xi _{n})_{n}\in \ell ^{2};\,|\xi _{n}|\leq {\tfrac {1}{n}}{\mbox{ für alle }}n\}\subset \ell ^{2},\quad \varphi ((x_{n})_{n\in \mathbb {N} }):=({\tfrac {2x_{n}-1}{n}})_{n\in \mathbb {N} }}$

is a homeomorphism.

• Every compact metric space is equally Eberlein-compact, because such spaces are homeomorphic to closed subspaces of the Hilbert cube.
• The unit sphere of the dual space of the sequence space with the weak - * - topology is a compact Hausdorff space according to the Banach-Alaoglu theorem . This room is not Eberlein-compact. ${\displaystyle \ell ^{\infty }}$

properties

• For Eberlein-compact spaces, the inferences from the Eberlein – Šmulian theorem apply , in particular such spaces are sequence- compact and a subset is closed if and only if it contains its limit value with every convergent sequence .
• For a compact Hausdorff space let the function space of the continuous functions with the supremum norm . Then the following statements are equivalent: ${\displaystyle \Omega }$ ${\displaystyle C(\Omega )}$ ${\displaystyle \Omega \rightarrow \mathbb {R} }$
  • ${\displaystyle \Omega }$ is Eberlein-compact.
  • ${\displaystyle C(\Omega )}$is a WCG room .
  • The unit sphere of the dual space with the weak - * - topology is Eberlein-compact. ${\displaystyle C(\Omega )'}$

Equivalent characterizations

Topological characterization

The definition of Eberlein compact space uses a Banach space. The following topological characterization, which has no reference to Banach spaces, goes back to Haskell Rosenthal :

A compact Hausdorff space is Eberlein-compact if and only if there is a sequence such that it holds ${\displaystyle \Omega }$ ${\displaystyle ({\mathcal {G}}_{n})_{n\in \mathbb {N} }}$

• Each is a family of open F _σ -sets ${\displaystyle {\mathcal {G}}_{n}}$
• For each, there are at most a finite number of with each, in short: each is pointless. ${\displaystyle \omega \in \Omega }$ ${\displaystyle n}$ ${\displaystyle G\in {\mathcal {G}}_{n}}$ ${\displaystyle \omega \in G}$ ${\displaystyle {\mathcal {G}}_{n}}$
• For everyone with there is one and one , so that , being the characteristic function of the set denotes. ${\displaystyle \omega _{1},\omega _{2}\in \Omega }$ ${\displaystyle \omega _{1}\not =\omega _{2}}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle G\in {\mathcal {G}}_{n}}$ ${\displaystyle \chi _{G}(\omega _{1})\not =\chi _{G}(\omega _{2})}$ ${\displaystyle \chi _{G}}$ ${\displaystyle G}$

Replacing the third condition with

• For everyone with there is one and one with and ${\displaystyle \omega _{1},\omega _{2}\in \Omega }$ ${\displaystyle \omega _{1}\not =\omega _{2}}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle G\in {\mathcal {G}}_{n}}$ ${\displaystyle \omega _{1}\in G}$ ${\displaystyle \omega _{2}\notin G}$

this gives a characterization of the metrisable Eberlein compact rooms.

Special Banach rooms

The same class of compact spaces is obtained if one restricts the Banach spaces used in the definition of Eberlein compactness. The following statements about a topological space are equivalent: ${\displaystyle \Omega }$

• ${\displaystyle \Omega }$ is Eberlein-compact.
• ${\displaystyle \Omega }$ is homeomorphic to a weakly compact subset of a reflexive Banach space in the relative weak topology.
• ${\displaystyle \Omega }$is homeomorphic to a weakly compact subset of a Banach space in the relative weak topology, where the Banach space ${\displaystyle c_{0}(\Gamma )}$ ${\displaystyle c_{0}(\Gamma )}$

${\displaystyle c_{0}(\Gamma ):=\{(x_{\gamma })_{\gamma \in \Gamma }|\,\{\gamma \in \Gamma |\,|x_{\gamma }|>\varepsilon \}{\text{ ist endlich für jedes }}\varepsilon >0\,\}}$

with the supremum norm is.

Some authors use the latter characterization as a definition.

Individual evidence

1. ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Springer-Verlag 1975, ISBN 3-540-07402-3 , Chapter 5, §2: Definition on page 146
2. ↑ K. Kunen, J. Vaughan: Handbook of Set-Theoretic Topology , Elsevier-Verlag 2014, Chapter 13, §6, Definition 6.2
3. ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Springer-Verlag 1975, ISBN 3-540-07402-3 , Chapter 5, §2, Theorem 4
4. ↑ HP Rosenthal: The hereditary problem for weakly compactly generated Banach spaces , Composito Math. (1974), Volume 28, pages 83-111
5. ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Springer-Verlag 1975, ISBN 3-540-07402-3 , Chapter 5, §3: Rosenthal's topological characterization of Eberlein compacts
6. ↑ Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos, Václav Zizler: Banach Space Theory: The Basis for Linear and Nonlinear Analysis , Springer Science & Business Media 2011, Chapter 14.1 Eberlein Compact Spaces
7. ↑ Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos, Václav Zizler: Banach Space Theory: The Basis for Linear and Nonlinear Analysis , Springer Science & Business Media 2011, Definition 3.18