Weakly compact generated space

Spaces that are generated in a weakly compact manner are examined in the mathematical subfield of functional analysis. It is about the large class of Banach spaces that are generated by a weakly compact set. The fundamental investigations into this room class go back to Joram Lindenstrauss . After the English term weakly compactly generated space , such rooms are also called WCG rooms .

definition

A Banach space is called weakly compact if it is produced by a weakly compact set, that is, there is a weakly compact set in this Banach space, so that the closed envelope of the subspace generated by already coincides with the total space. ${\displaystyle K}$ ${\displaystyle K}$

Examples

• Every separable Banach space is generated weakly compact. Namely, if a dense subset , then there is even a norm- compact generating system. ${\displaystyle \{x_{n}|\,n\in \mathbb {N} \}}$ ${\displaystyle \{{\frac {x_{n}}{n\|x_{n}\|}}|\,n\in \mathbb {N} ,x_{n}\not =0\}\cup \{0\}}$
• Every reflexive Banach space is generated weakly compact, because one of the equivalent characterizations of reflexivity is that the unit sphere is weakly compact, and this naturally creates the Banach space, even without additional closure formation.
• Is the separable sequence space of the null sequences with the supremum norm and ${\displaystyle c_{0}}$

${\displaystyle \ell ^{2}(\mathbb {R} ):=\{(x_{r})_{r\in \mathbb {R} }|\,\sum _{r\in \mathbb {R} }|x_{r}|^{2}<\infty \}}$

the non-separable Hilbert space with the 2-norm , the direct sum is generated weakly compact, but neither separable nor reflexive. ${\displaystyle c_{0}\oplus \ell ^{2}(\mathbb {R} )}$

• For a compact Hausdorff space , the Banach space of continuous functions with the supremum norm is created weakly compact if and only if Eberlein is compact . ${\displaystyle \Omega }$ ${\displaystyle C(\Omega )}$ ${\displaystyle \Omega \rightarrow \mathbb {R} }$ ${\displaystyle \Omega }$
• ${\displaystyle \ell ^{1}(I):=\{(x_{i})_{i\in I}|\,\sum _{i\in I}|x_{i}|<\infty \}}$is generated weakly compact if and only if the index set is countable . ${\displaystyle I}$

properties

• Let be a weakly compact generated Banach space. Then there is a set and an injective, continuous, linear operator , where ${\displaystyle X}$ ${\displaystyle \Gamma }$ ${\displaystyle X\rightarrow c_{0}(\Gamma )}$

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

• The following theorem by Davis , Figiel , Johnson , Pełczyński shows the proximity of the weakly compact generated spaces to reflexive spaces. A Banach space is created weakly compact if and only if there is a reflexive space and an injective, continuous, linear operator with a dense image. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Y\rightarrow X}$
• According to Troyanski's theorem, spaces that are generated weakly compact have an equivalent norm which makes the space a locally uniformly convex space ; with this equivalent norm one can even achieve that additionally smooth and the dual space norm is strictly convex . ${\displaystyle X}$
• A closed subspace of a Banach space is called quasi-complemented if there is a closed subspace such that and . If a Banach space is generated in a weakly compact manner, then every closed subspace is quasi-complemented according to a Lindenstrauss theorem . ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle Z\subset X}$ ${\displaystyle Y\cap Z=\{0\}}$ ${\displaystyle {\overline {Y+Z}}=X}$ ${\displaystyle X}$
• According to the Banach-Alaoglu theorem, the unit sphere of the dual space of a normalized space is compact in the weak - * - topology . Amir-Lindenstrauss's theorem says that the unit sphere of the dual space of a weakly compact generated Banach space in the weakly - * - topology is also sequentially compact .

Individual evidence

1. ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Springer-Verlag 1975, ISBN 3-540-07402-3 , chapter 5, Weakly Compactly Generated Banach Spaces
2. ↑ Marian Fabian, Petr Habala, Petr Hajek, Vicente Montesinos Santalucia, Jan Pelant, Vaclav Zizler: Functional Analysis and Infinite-Dimensional Geometry , CMS Books in Mathematics, Springer Science & Business Media 2013, ISBN 1-475-73480-8 , chapter 11: Weakly Compactly Generated Spaces
3. ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Springer-Verlag 1975, ISBN 3-540-07402-3 , Chapter 5, §2, Theorem 4
4. ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Springer-Verlag 1975, ISBN 3-540-07402-3 , Chapter 5, §2, Theorem 1
5. ↑ Marian Fabian, Petr Habala, Petr Hajek, Vicente Montesinos Santalucia, Jan Pelant, Vaclav Zizler: Functional Analysis and Infinite-Dimensional Geometry , CMS Books in Mathematics, Springer Science & Business Media 2013, ISBN 1-475-73480-8 , corollary 11.13
6. ↑ D. Amir and J. Lindenstrauss: The Structure of Weakly Compact Sets in Banach Spaces , Annals of Mathematics, Volume 88, No. 1 (1968), pp. 35-46, Main Theorem
7. ↑ Marian Fabian, Petr Habala, Petr Hajek, Vicente Montesinos Santalucia, Jan Pelant, Vaclav Zizler: Functional Analysis and Infinite-Dimensional Geometry , CMS Books in Mathematics, Springer Science & Business Media 2013, ISBN 1-475-73480-8 , Theorem 11.17 + Corollary 11.19
8. ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Springer-Verlag 1975, ISBN 3-540-07402-3 , Chapter 5, §5: Trojanski's Theorem
9. ^ S. Troyanski: On locally uniformly convex and differentiable norms in certain nonseparable Banach spaces , Studia Mathematica 1972, volume 43, pages 125-138
10. ↑ Marian Fabian, Petr Habala, Petr Hajek, Vicente Montesinos Santalucia, Jan Pelant, Vaclav Zizler: Functional Analysis and Infinite-Dimensional Geometry , CMS Books in Mathematics, Springer Science & Business Media 2013, ISBN 1-475-73480-8 , Theorem 11.41
11. ↑ Terry J. Morrison: Functional Analysis: An Introduction to Banach Space Theory , John Wiley & Sons 2011, ISBN 1-118-03124-5 , Theorem 4.8