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Operators whose dual has non-separable range
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TL;DR: In this paper, the authors characterize the non-separability of a bounded linear operator by fixing properties of the operator, and show that the operator is not separable in the Banach space.
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Abstract: Let $X$ and $Y$ be separable Banach spaces and $T:X\to Y$ be a bounded linear operator. We characterize the non-separability of $T^*(Y^*)$ by means of fixing properties of the operator $T$.
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Finite basis for analytic multiple gaps
TL;DR: In this paper, it was shown that for every positive integer n there is a finite basis for the class of analytic n-gaps, i.e., an n-gap consists of n many pairwise orthogonal families of subsets of a countable set that cannot be separated.
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A density version of the Halpern–Läuchli theorem
TL;DR: A density version of the Halpern-Lauchli Theorem has been proved in this paper, where it is shown that a tree T is homogeneous if T has a unique root and there exists an integer b ⩾ 2 such that every t ∈ T has exactly b immediate successors.
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Tukey classification of some ideals on ω and the lattices of weakly compact sets in Banach spaces
TL;DR: In this paper, the lattice structure of weakly compact subsets of the unit ball B X of a separable Banach space X, equipped with the inclusion relation (this structure is denoted by K ( B X ) ) and also with the parametrized family of almost inclusion relations K ⊆ L + e B X, where e > 0, is studied.
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Banach spaces and Ramsey Theory: some open problems
TL;DR: In this paper, the authors discuss some open problems in the Geometry of Banach spaces having Ramsey-theoretic flavor, together with well known results related to them, and expose the problems with known results.
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Dense subsets of products of finite trees
TL;DR: In this paper, a "uniform" version of the finite density Halpern-Lauchli Theorem was shown to be true for homogeneous trees, where every tree is homogeneous if it is uniquely rooted and there is an integer called the branching number of the tree, such that every tree has exactly the same immediate successors.
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References
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Classical descriptive set theory
Alexander S. Kechris
- 01 Jan 1987
TL;DR: In this article, the authors present a largely balanced approach, which combines many elements of the different traditions of the subject, and includes a wide variety of examples, exercises, and applications, in order to illustrate the general concepts and results of the theory.
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Handbook of the Geometry of Banach spaces
William B. Johnson,Joram Lindenstrauss +1 more
- 01 Jan 2001
TL;DR: Banach spaces have been studied extensively in the literature, see as mentioned in this paper for a survey of some of the main aspects of the Banach spaces and its application in the analysis of finite dimensional subspaces.
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A double-dual characterization of separable Banach spaces containingl 1
TL;DR: In this paper, it was shown that a separable Banach space B contains a subspace isomorphic to the double-dual of B if and only if there exists an element inB**, which is not a weak* limit of a sequence of elements in B. Consequently, B contains an isomorph of B** if (and only if) the cardinality ofB** is greater than that of the continuum.
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