On Weakly-Compactly Generated Asplund Banach spaces

2004 ◽  
Vol 55 (1) ◽  
pp. 77-85 ◽  
Author(s):  
L. Oncina
Keyword(s):  
Banach Spaces ◽  
Compactly Generated

scholarly journals Schauder decompositions in non-separable Banach spaces

1972 ◽  
Vol 6 (1) ◽  
pp. 133-144 ◽  
Author(s):  
J.J.M. Chadwick ◽  
R.W. Cross
Keyword(s):  
Banach Spaces ◽  
Separable Spaces ◽  
Compactly Generated

It is shown that Schauder decompositions exist in non-separable weakly compactly generated spaces and in certain non-separable conjugate spaces. Some results are obtained concerning shrinking and boundedly complete Schauder decompositions in non-separable spaces.

scholarly journals Twisted properties of banach spaces

2001 ◽  
Vol 89 (2) ◽  
pp. 217 ◽  
Author(s):  
Jesús M. F. Castillo ◽  
Manuel Gonzáles ◽  
Anatolij M. Plichko ◽  
David Yost
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Topological Properties ◽  
Space Property ◽  
Compactly Generated

If $\mathcal P$, $\mathcal Q$ are two linear topological properties, say that a Banach space $X$ has the property $\mathcal P$-by-$\mathcal Q$ (or is a $\mathcal P$-by-$\mathcal Q$ space) if $X$ has a subspace $Y$ with property $\mathcal P$ such that the corresponding quotient $X/Y$ has property $\mathcal Q$. The choices $\mathcal P,\mathcal Q \in\{\hbox{separable, reflexive}\}$ lead naturally to some new results and new proofs of old results concerning weakly compactly generated Banach spaces. For example, every extension of a subspace of $L_1(0,1)$ by a WCG space is WCG. They also give a simple new example of a Banach space property which is not a 3-space property but whose dual is a 3-space property.

scholarly journals Inner characterizations of weakly compactly generated Banach spaces and their relatives

2004 ◽  
Vol 297 (2) ◽  
pp. 419-455 ◽  
Author(s):  
M. Fabian ◽  
G. Godefroy ◽  
V. Montesinos ◽  
V. Zizler
Keyword(s):  
Banach Spaces ◽  
Compactly Generated

COMPLEMENTATION AND EMBEDDINGS OF c0(I) IN BANACH SPACES

2002 ◽  
Vol 85 (3) ◽  
pp. 742-768 ◽  
Author(s):  
SPIROS A. ARGYROS ◽  
JESÚS F. CASTILLO ◽  
ANTONIO S. GRANERO ◽  
MAR JIMÉNEZ ◽  
JOSÉ P. MORENO
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Compact Subset ◽  
Continuum Hypothesis ◽  
Finite Partition ◽  
Weakly Compact ◽  
Complemented Subspaces ◽  
Generalized Continuum ◽  
Compactly Generated

We investigate in this paper the complementation of copies of $c_0(I)$ in some classes of Banach spaces (in the class of weakly compactly generated (WCG) Banach spaces, in the larger class $\mathcal{V}$ of Banach spaces which are subspaces of some $C(K)$ space with $K$ a Valdivia compact, and in the Banach spaces $C([1, \alpha ])$, where $\alpha$ is an ordinal) and the embedding of $c_0(I)$ in the elements of the class $\mathcal{C}$ of complemented subspaces of $C(K)$ spaces. Two of our results are as follows:(i) in a Banach space $X \in \mathcal{V}$ every copy of $c_0(I)$ with $\# I < \aleph _{\omega}$ is complemented;(ii) if $\alpha _0 = \aleph _0$, $\alpha _{n+1} = 2^{\alpha _n}$, $n \geq 0$, and $\alpha = \sup \{\alpha _n : n \geq 0\}$ there exists a WCG Banach space with an uncomplemented copy of $c_0(\alpha )$.So, under the generalized continuum hypothesis (GCH), $\aleph _{\omega}$ is the greatest cardinal $\tau$ such that every copy of $c_0(I)$ with $\# I < \tau$ is complemented in the class $\mathcal{V}$. If $T : c_0(I) \to C([1,\alpha ])$ is an isomorphism into its image, we prove that:(i) $c_0(I)$ is complemented, whenever $\| T \| ,\| T^{-1} \| < (3/2)^{\frac 12}$;(ii) there is a finite partition $\{I_1, \dots , I_k\}$ of $I$ such that each copy $T(c_0(I_k))$ is complemented.Concerning the class $\mathcal{C}$, we prove that an already known property of $C(K)$ spaces is still true for this class, namely, if $X \in \mathcal{C}$, the following are equivalent:(i) there is a weakly compact subset $W \subset X$ with ${\rm Dens}(W) = \tau$;(ii) $c_0(\tau )$ is isomorphically embedded into $X$.This yields a new characterization of a class of injective Banach spaces.2000 Mathematical Subject Classification: 46B20, 46B26.

A CHARACTERIZATION OF SUBSPACES OF WEAKLY COMPACTLY GENERATED BANACH SPACES

2004 ◽  
Vol 69 (02) ◽  
pp. 457-464 ◽  
Author(s):  
M. FABIAN ◽  
V. MONTESINOS ◽  
V. ZIZLER
Keyword(s):  
Banach Spaces ◽  
Compactly Generated

FRAMES OF SUBSPACES FOR BANACH SPACES

2010 ◽  
Vol 08 (02) ◽  
pp. 243-252 ◽  
Author(s):  
P. K. JAIN ◽  
S. K. KAUSHIK ◽  
VARINDER KUMAR
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Banach Frame ◽  
Necessary And Sufficient ◽  
Compactly Generated

Frames of subspaces for Banach spaces have been introduced and studied. Examples and counter-examples to distinguish various types of frames of subspaces have been given. It has been proved that if a Banach space has a Banach frame, then it also has a frame of subspaces. Also, a necessary and sufficient condition for a sequence of projections, associated with a frame of subspaces, to be unique has been given. Finally, we consider complete frame of subspaces and prove that every weakly compactly generated Banach space has a complete frame of subspaces.

scholarly journals UNIVERSAL SUBGROUPS OF POLISH GROUPS

2014 ◽  
Vol 79 (4) ◽  
pp. 1148-1183 ◽  
Author(s):  
KONSTANTINOS A. BEROS
Keyword(s):  
Topological Group ◽  
Banach Spaces ◽  
Topological Groups ◽  
Locally Compact ◽  
Polish Groups ◽  
Countable Power ◽  
Compact Case ◽  
Image Position ◽  
The Relationship ◽  
Compactly Generated

AbstractGiven a class${\cal C}$of subgroups of a topological groupG, we say that a subgroup$H \in {\cal C}$is auniversal${\cal C}$subgroupofGif every subgroup$K \in {\cal C}$is a continuous homomorphic preimage ofH. Such subgroups may be regarded as complete members of${\cal C}$with respect to a natural preorder on the set of subgroups ofG. We show that for any locally compact Polish groupG, the countable powerGωhas a universalKσsubgroup and a universal compactly generated subgroup. We prove a weaker version of this in the nonlocally compact case and provide an example showing that this result cannot readily be improved. Additionally, we show that many standard Banach spaces (viewed as additive topological groups) have universalKσand compactly generated subgroups. As an aside, we explore the relationship between the classes ofKσand compactly generated subgroups and give conditions under which the two coincide.

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