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1
Content available remote

Constructing non-compact operators into c₀

100%
Banakh I., Banakh T.
Studia Mathematica
|
2010
|
tom 201
|
nr 1
65-67
EN
We prove that for each dense non-compact linear operator S: X → Y between Banach spaces there is a linear operator T: Y → c₀ such that the operator TS: X → c₀ is not compact. This generalizes the Josefson-Nissenzweig Theorem.
2
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Topological Structure of Non-separable Sigma-locally Compact Convex Sets

81%
Banakh I., Banakh T., Koshino K.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2013
|
tom 61
|
nr 2
149-153
3
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On local convexity of nonlinear mappings between Banach spaces

81%
Banakh I., Banakh T., Plichko A., Prykarpatsky A.
Open Mathematics
|
2012
|
tom 10
|
nr 6
2264-2271
EN
We find conditions for a smooth nonlinear map f: U → V between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some c and each positive ɛ < c the image f(B ɛ(x)) of each ɛ-ball B ɛ(x) ⊂ U is convex. We give a lower bound on c via the second order Lipschitz constant Lip2(f), the Lipschitz-open constant Lipo(f) of f, and the 2-convexity number conv2(X) of the Banach space X.
4
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Extenders for vector-valued functions

81%
Banakh I., Banakh T., Yamazaki K.
Studia Mathematica
|
2009
|
tom 191
|
nr 2
123-150
EN
Given a subset A of a topological space X, a locally convex space Y, and a family ℂ of subsets of Y we study the problem of the existence of a linear ℂ-extender $u: C_{∞}(A,Y) → C_{∞}(X,Y)$, which is a linear operator extending bounded continuous functions f: A → C ⊂ Y, C ∈ ℂ, to bounded continuous functions f̅ = u(f): X → C ⊂ Y. Two necessary conditions for the existence of such an extender are found in terms of a topological game, which is a modification of the classical strong Choquet game. The results obtained allow us to characterize reflexive Banach spaces as the only normed spaces Y such that for every closed subset A of a GO-space X there is a ℂ-extender $u: C_{∞}(A,Y) → C_{∞}(X,Y)$ for the family ℂ of closed convex subsets of Y. Also we obtain a characterization of Polish spaces and of weakly sequentially complete Banach lattices in terms of extenders.
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