Wyniki wyszukiwania

1

Universal stability of Banach spaces for ε -isometries

100%

Cheng L., Dai D., Dong Y., Zhou Y.

Studia Mathematica

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2014

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tom 221

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nr 2

141-149

EN

Let X, Y be real Banach spaces and ε > 0. A standard ε-isometry f: X → Y is said to be (α,γ)-stable (with respect to $T: L(f) ≡ \overline{span}f(X) → X$ for some α,γ > 0) if T is a linear operator with ||T|| ≤ α such that Tf- Id is uniformly bounded by γε on X. The pair (X,Y) is said to be stable if every standard ε-isometry f: X → Y is (α,γ)-stable for some α,γ > 0. The space X[Y] is said to be universally left [right]-stable if (X,Y) is always stable for every Y[X]. In this paper, we show that universally right-stable spaces are just Hilbert spaces; every injective space is universally left-stable; a Banach space X isomorphic to a subspace of $ℓ_{∞}$ is universally left-stable if and only if it is isomorphic to $ℓ_{∞}$; and a separable space X has the property that (X,Y) is left-stable for every separable Y if and only if X is isomorphic to c₀.

2

On some new characterizations of weakly compact sets in Banach spaces

100%

Cheng L., Cheng Q., Luo Z.

Studia Mathematica

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2010

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tom 201

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nr 2

155-166

EN

We show several characterizations of weakly compact sets in Banach spaces. Given a bounded closed convex set C of a Banach space X, the following statements are equivalent: (i) C is weakly compact; (ii) C can be affinely uniformly embedded into a reflexive Banach space; (iii) there exists an equivalent norm on X which has the w2R-property on C; (iv) there is a continuous and w*-lower semicontinuous seminorm p on the dual X* with $p ≥ sup_{C}$ such that p² is everywhere Fréchet differentiable in X*; and as a consequence, the space X is a weakly compactly generated space if and only if there exists a continuous and w*-l.s.c. Fréchet smooth (not necessarily equivalent) norm on X*.

3

Minimal ball-coverings in Banach spaces and their application

100%

Cheng L., Cheng Q., Shi H.

Studia Mathematica

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2009

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tom 192

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nr 1

15-27

EN

By a ball-covering 𝓑 of a Banach space X, we mean a collection of open balls off the origin in X and whose union contains the unit sphere of X; a ball-covering 𝓑 is called minimal if its cardinality $𝓑^{#}$ is smallest among all ball-coverings of X. This article, through establishing a characterization for existence of a ball-covering in Banach spaces, shows that for every n ∈ ℕ with k ≤ n there exists an n-dimensional space admitting a minimal ball-covering of n + k balls. As an application, we give a new characterization of superreflexive spaces in terms of ball-coverings. Finally, we show that every infinite-dimensional Banach space admits an equivalent norm such that there is an infinite-dimensional quotient space possessing a countable ball-covering.

4

On super-weakly compact sets and uniformly convexifiable sets

100%

Cheng L., Cheng Q., Wang B., Zhang W.

Studia Mathematica

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2010

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tom 199

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nr 2

145-169

EN

This paper mainly concerns the topological nature of uniformly convexifiable sets in general Banach spaces: A sufficient and necessary condition for a bounded closed convex set C of a Banach space X to be uniformly convexifiable (i.e. there exists an equivalent norm on X which is uniformly convex on C) is that the set C is super-weakly compact, which is defined using a generalization of finite representability. The proofs use appropriate versions of classical theorems, such as James' finite tree theorem, Enflo's renorming technique, Grothendieck's lemma and the Davis-Figiel-Johnson-Pełczyński lemma.

Ograniczanie wyników

4 Studia Mathematica

4 Cheng L.

3 Cheng Q.

1 Dai D.

1 Dong Y.

1 Luo Z.

1 Shi H.

1 Wang B.

1 Zhang W.

1 Zhou Y.

1 2014

2 2010

1 2009

Universal stability of Banach spaces for ε -isometries

On some new characterizations of weakly compact sets in Banach spaces

Minimal ball-coverings in Banach spaces and their application

On super-weakly compact sets and uniformly convexifiable sets