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C(K) spaces which cannot be uniformly embedded into c₀(Γ)

Jan Pelant, Petr Holický, Ondřej F. K. Kalenda (2006)

Fundamenta Mathematicae

We give two examples of scattered compact spaces K such that C(K) is not uniformly homeomorphic to any subset of c₀(Γ) for any set Γ. The first one is [0,ω₁] and hence it has the smallest possible cardinality, the other one has the smallest possible height ω₀ + 1.

Complex Banach spaces with Valdivia dual unit ball.

Ondrej F. K. Kalenda (2005)

Extracta Mathematicae

We study the classes of complex Banach spaces with Valdivia dual unit ball. We give complex analogues of several theorems on real spaces. Further we study relationship of these complex Banach spaces with their real versions and that of real Banach spaces and their complexification. We also formulate several open problems.

Continuity properties up to a countable partition.

Aníbal Moltó, José Orihuela, Stanimir Troyanski, Manuel Valdivia (2006)

RACSAM

Approximation and rigidity properties in renorming constructions are characterized with some classes of simple maps. Those maps describe continuity properties up to a countable partition. The construction of such kind of maps can be done with ideas from the First Lebesgue Theorem. We present new results on the relationship between Kadec and locally uniformly rotund renormability as well as characterizations of the last one with the simple maps used here.

Countable products of spaces of finite sets

Antonio Avilés (2005)

Fundamenta Mathematicae

We consider the compact spaces σₙ(Γ) of subsets of Γ of cardinality at most n and their countable products. We give a complete classification of their Banach spaces of continuous functions and a partial topological classification.

Decomposition of Banach Space into a Direct Sum of Separable and Reflexive Subspaces and Borel Maps

Plichko, Anatolij (1997)

Serdica Mathematical Journal

* This paper was supported in part by the Bulgarian Ministry of Education, Science and Technologies under contract MM-506/95.The main results of the paper are: Theorem 1. Let a Banach space E be decomposed into a direct sum of separable and reflexive subspaces. Then for every Hausdorff locally convex topological vector space Z and for every linear continuous bijective operator T : E → Z, the inverse T^(−1) is a Borel map. Theorem 2. Let us assume the continuum hypothesis. If a Banach space E cannot...

Descriptive compact spaces and renorming

L. Oncina, M. Raja (2004)

Studia Mathematica

We study the class of descriptive compact spaces, the Banach spaces generated by descriptive compact subsets and their relation to renorming problems.

Distances to convex sets

Antonio S. Granero, Marcos Sánchez (2007)

Studia Mathematica

If X is a Banach space and C a convex subset of X*, we investigate whether the distance d ̂ ( c o ¯ w * ( K ) , C ) : = s u p i n f | | k - c | | : c C : k c o ¯ w * ( K ) from c o ¯ w * ( K ) to C is M-controlled by the distance d̂(K,C) (that is, if d ̂ ( c o ¯ w * ( K ) , C ) M d ̂ ( K , C ) for some 1 ≤ M < ∞), when K is any weak*-compact subset of X*. We prove, for example, that: (i) C has 3-control if C contains no copy of the basis of ℓ₁(c); (ii) C has 1-control when C ⊂ Y ⊂ X* and Y is a subspace with weak*-angelic closed dual unit ball B(Y*); (iii) if C is a convex subset of X and X is considered canonically embedded into...

Equilateral sets in Banach spaces of the form C(K)

Sophocles K. Mercourakis, Georgios Vassiliadis (2015)

Studia Mathematica

We show that for "most" compact nonmetrizable spaces, the unit ball of the Banach space C(K) contains an uncountable 2-equilateral set. We also give examples of compact nonmetrizable spaces K such that the minimum cardinality of a maximal equilateral set in C(K) is countable.

Extension operators on balls and on spaces of finite sets

Antonio Avilés, Witold Marciszewski (2015)

Studia Mathematica

We study extension operators between spaces of continuous functions on the spaces σ ( 2 X ) of subsets of X of cardinality at most n. As an application, we show that if B H is the unit ball of a nonseparable Hilbert space H equipped with the weak topology, then, for any 0 < λ < μ, there is no extension operator T : C ( λ B H ) C ( μ B H ) .

Fixed point free maps of a closed ball with small measures of noncompactness.

Martin Väth (2001)

Collectanea Mathematica

We show that in all infinite-dimensional normed spaces it is possible to construct a fixed point free continuous map of the unit ball whose measure of noncompactness is bounded by 2. Moreover, for a large class of spaces (containing separable spaces, Hilbert spaces and l-infinity (S)) even the best possible bound 1 is attained for certain measures of noncompactness.

Fragmentability and compactness in C(K)-spaces

B. Cascales, G. Manjabacas, G. Vera (1998)

Studia Mathematica

Let K be a compact Hausdorff space, C p ( K ) the space of continuous functions on K endowed with the pointwise convergence topology, D ⊂ K a dense subset and t p ( D ) the topology in C(K) of pointwise convergence on D. It is proved that when C p ( K ) is Lindelöf the t p ( D ) -compact subsets of C(K) are fragmented by the supremum norm of C(K). As a consequence we obtain some Namioka type results and apply them to prove that if K is separable and C p ( K ) is Lindelöf, then K is metrizable if, and only if, there is a countable and dense...

Fragmentability and σ-fragmentability

J. Jayne, I. Namioka, C. Rogers (1993)

Fundamenta Mathematicae

Recent work has studied the fragmentability and σ-fragmentability properties of Banach spaces. Here examples are given that justify the definitions that have been used. The fragmentability and σ-fragmentability properties of the spaces and c ( Γ ) , with Γ uncountable, are determined.

(I)-envelopes of unit balls and James' characterization of reflexivity

Ondřej F. K. Kalenda (2007)

Studia Mathematica

We study the (I)-envelopes of the unit balls of Banach spaces. We show, in particular, that any nonreflexive space can be renormed in such a way that the (I)-envelope of the unit ball is not the whole bidual unit ball. Further, we give a simpler proof of James' characterization of reflexivity in the nonseparable case. We also study the spaces in which the (I)-envelope of the unit ball adds nothing.

Isometric embeddings of a class of separable metric spaces into Banach spaces

Sophocles K. Mercourakis, Vassiliadis G. Vassiliadis (2018)

Commentationes Mathematicae Universitatis Carolinae

Let ( M , d ) be a bounded countable metric space and c > 0 a constant, such that d ( x , y ) + d ( y , z ) - d ( x , z ) c , for any pairwise distinct points x , y , z of M . For such metric spaces we prove that they can be isometrically embedded into any Banach space containing an isomorphic copy of .

Kadec norms and Borel sets in a Banach space

M. Raja (1999)

Studia Mathematica

We introduce a property for a couple of topologies that allows us to give simple proofs of some classic results about Borel sets in Banach spaces by Edgar, Schachermayer and Talagrand as well as some new results. We characterize the existence of Kadec type renormings in the spirit of the new results for LUR spaces by Moltó, Orihuela and Troyanski.

Kadec Norms on Spaces of Continuous Functions

Burke, Maxim R., Wiesaw, Kubis, Stevo, Todorcevic (2006)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: Primary: 46B03, 46B26. Secondary: 46E15, 54C35.We study the existence of pointwise Kadec renormings for Banach spaces of the form C(K). We show in particular that such a renorming exists when K is any product of compact linearly ordered spaces, extending the result for a single factor due to Haydon, Jayne, Namioka and Rogers. We show that if C(K1) has a pointwise Kadec renorming and K2 belongs to the class of spaces obtained by closing the class of compact...

Lipschitz and uniform embeddings into

N. J. Kalton (2011)

Fundamenta Mathematicae

We show that there is no uniformly continuous selection of the quotient map Q : / c relative to the unit ball. We use this to construct an answer to a problem of Benyamini and Lindenstrauss; there is a Banach space X such that there is a no Lipschitz retraction of X** onto X; in fact there is no uniformly continuous retraction from B X * * onto B X .

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