# 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 .${\ displaystyle [0,1] ^ {\ infty}}$ ${\ displaystyle \ ell ^ {2}}$ ${\ displaystyle \ varphi: [0,1] ^ {\ infty} \ rightarrow \ {(\ xi _ {n}) _ {n} \ in \ ell ^ {2}; \, | \ xi _ {n} | \ leq {\ tfrac {1} {n}} {\ mbox {for all}} n \} \ subset \ ell ^ {2}, \ quad \ varphi ((x_ {n}) _ {n \ in \ mathbb {N}}): = ({\ tfrac {2x_ {n} -1} {n}}) _ {n \ in \ mathbb {N}}}$ 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${\ displaystyle c_ {0} (\ Gamma)}$ ${\ displaystyle c_ {0} (\ Gamma)}$ ${\ displaystyle c_ {0} (\ Gamma): = \ {(x _ {\ gamma}) _ {\ gamma \ in \ Gamma} | \, \ {\ gamma \ in \ Gamma | \, | x _ {\ gamma } |> \ varepsilon \} {\ text {is finite for every}} \ varepsilon> 0 \, \}}$ 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