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Limit points of arithmetic means of sequences in Banach spaces

Roman Lávička (2000)

Commentationes Mathematicae Universitatis Carolinae

We shall prove the following statements: Given a sequence { a n } n = 1 ∞ in a Banach space 𝐗 enjoying the weak Banach-Saks property, there is a subsequence (or a permutation) { b n } n = 1 ∞ of the sequence { a n } n = 1 ∞ such that lim n → ∞ 1 n ∑ j = 1 n b j = a whenever a belongs to the closed convex hull of the set of weak limit points of { a n } n = 1 ∞ . In case 𝐗 has the Banach-Saks property and { a n } n = 1 ∞ is bounded the converse assertion holds too. A characterization of reflexive spaces in terms of limit points and cores of bounded sequences is also given. The motivation for the...

Lineární posloupnosti

Miroslav Laitoch (1968)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica-Physica-Chemica

Measure of noncompactness of linear operators between spaces of sequences that are ( N ¯ , q ) summable or bounded

Eberhard Malkowsky, V. Rakočević (2001)

Czechoslovak Mathematical Journal

In this paper we investigate linear operators between arbitrary BK spaces X and spaces Y of sequences that are ( N ¯ , q ) summable or bounded. We give necessary and sufficient conditions for infinite matrices A to map X into Y . Further, the Hausdorff measure of noncompactness is applied to give necessary and sufficient conditions for A to be a compact operator.

Nearness relations in linear spaces

Martin Kalina (2004)

Kybernetika

In this paper, we consider nearness-based convergence in a linear space, where the coordinatewise given nearness relations are aggregated using weighted pseudo-arithmetic and geometric means and using continuous t-norms.

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