Wyniki wyszukiwania

1

Countably convex $G_{δ}$ sets

100%

Fonf V., Kojman M.

Fundamenta Mathematicae

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2001

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tom 168

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nr 2

131-140

EN

We investigate countably convex $G_{δ}$ subsets of Banach spaces. A subset of a linear space is countably convex if it can be represented as a countable union of convex sets. A known sufficient condition for countable convexity of an arbitrary subset of a separable normed space is that it does not contain a semi-clique [9]. A semi-clique in a set S is a subset P ⊆ S so that for every x ∈ P and open neighborhood u of x there exists a finite set X ⊆ P ∩ u such that conv(X) ⊈ S. For closed sets this condition is also necessary. We show that for countably convex $G_{δ}$ subsets of infinite-dimensional Banach spaces there are no necessary limitations on cliques and semi-cliques. Various necessary conditions on cliques and semi-cliques are obtained for countably convex $G_{δ}$ subsets of finite-dimensional spaces. The results distinguish dimension d ≤ 3 from dimension d ≥ 4: in a countably convex $G_{δ}$ subset of ℝ³ all cliques are scattered, whereas in ℝ⁴ a countably convex $G_{δ}$ set may contain a dense-in-itself clique.

2

Analytic and $C^k$ approximations of norms in separable Banach spaces

81%

Deville R., Fonf V., Hájek P.

Studia Mathematica

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1996

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tom 120

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nr 1

61-74

EN

We prove that in separable Hilbert spaces, in $ℓ_{p}(ℕ)$ for p an even integer, and in $L_{p}[0,1]$ for p an even integer, every equivalent norm can be approximated uniformly on bounded sets by analytic norms. In $ℓ_{p}(ℕ)$ and in $L_{p}[0,1]$ for p ∉ ℕ (resp. for p an odd integer), every equivalent norm can be approximated uniformly on bounded sets by $C^[p]}$-smooth norms (resp. by $C^{p-1}$-smooth norms).

3

On the uniform ergodic theorem in Banach spaces that do not contain duals

81%

Fonf V., Lin M., Rubinov A.

Studia Mathematica

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1996

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tom 121

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nr 1

67-85

EN

Let T be a power-bounded linear operator in a real Banach space X. We study the equality (*) $(I-T)X = {z ∈ X: sup_{n} ∥∑_{k=0}^{n} T^{k}z∥ < ∞}$. For X separable, we show that if T satisfies and is not uniformly ergodic, then $\overline{(I-T)X}$ contains an isomorphic copy of an infinite-dimensional dual Banach space. Consequently, if X is separable and does not contain isomorphic copies of infinite-dimensional dual Banach spaces, then (*) is equivalent to uniform ergodicity. As an application, sufficient conditions for uniform ergodicity of irreducible Markov chains on the (positive) integers are obtained.

Ograniczanie wyników

2 Studia Mathematica

1 Fundamenta Mathematicae

3 Fonf V.

1 Deville R.

1 Hájek P.

1 Kojman M.

1 Lin M.

1 Rubinov A.

1 2001

2 1996

Countably convex $G_{δ}$ sets

Analytic and $C^k$ approximations of norms in separable Banach spaces

On the uniform ergodic theorem in Banach spaces that do not contain duals