Limit points of arithmetic means of sequences in Banach spaces

Roman Lávička

Commentationes Mathematicae Universitatis Carolinae (2000)

  • Volume: 41, Issue: 1, page 97-106
  • ISSN: 0010-2628

Abstract

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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 problems investigated goes back to Lévy laplacian from potential theory in Hilbert spaces.

How to cite

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Lávička, Roman. "Limit points of arithmetic means of sequences in Banach spaces." Commentationes Mathematicae Universitatis Carolinae 41.1 (2000): 97-106. <http://eudml.org/doc/248599>.

@article{Lávička2000,
abstract = {We shall prove the following statements: Given a sequence $\lbrace a_n\rbrace _\{n=1\}^\{\infty \}$ in a Banach space $\mathbf \{X\}$ enjoying the weak Banach-Saks property, there is a subsequence (or a permutation) $\lbrace b_n\rbrace _\{n=1\}^\{\infty \}$ of the sequence $\lbrace a_n\rbrace _\{n=1\}^\{\infty \}$ such that \[ \lim \_\{n\rightarrow \infty \} \{1\over n\}\sum \_\{j=1\}^n b\_j=a \] whenever $a$ belongs to the closed convex hull of the set of weak limit points of $\lbrace a_n\rbrace _\{n=1\}^\{\infty \}$. In case $\mathbf \{X\}$ has the Banach-Saks property and $\lbrace a_n\rbrace _\{n=1\}^\{\infty \}$ 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 problems investigated goes back to Lévy laplacian from potential theory in Hilbert spaces.},
author = {Lávička, Roman},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Banach-Saks property; arithmetic means; limit points; subsequences; permutations of sequences; Banach-Saks property; arithmetic means; permutations of sequences; limit points; cores of sequences},
language = {eng},
number = {1},
pages = {97-106},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Limit points of arithmetic means of sequences in Banach spaces},
url = {http://eudml.org/doc/248599},
volume = {41},
year = {2000},
}

TY - JOUR
AU - Lávička, Roman
TI - Limit points of arithmetic means of sequences in Banach spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2000
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 41
IS - 1
SP - 97
EP - 106
AB - We shall prove the following statements: Given a sequence $\lbrace a_n\rbrace _{n=1}^{\infty }$ in a Banach space $\mathbf {X}$ enjoying the weak Banach-Saks property, there is a subsequence (or a permutation) $\lbrace b_n\rbrace _{n=1}^{\infty }$ of the sequence $\lbrace a_n\rbrace _{n=1}^{\infty }$ such that \[ \lim _{n\rightarrow \infty } {1\over n}\sum _{j=1}^n b_j=a \] whenever $a$ belongs to the closed convex hull of the set of weak limit points of $\lbrace a_n\rbrace _{n=1}^{\infty }$. In case $\mathbf {X}$ has the Banach-Saks property and $\lbrace a_n\rbrace _{n=1}^{\infty }$ 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 problems investigated goes back to Lévy laplacian from potential theory in Hilbert spaces.
LA - eng
KW - Banach-Saks property; arithmetic means; limit points; subsequences; permutations of sequences; Banach-Saks property; arithmetic means; permutations of sequences; limit points; cores of sequences
UR - http://eudml.org/doc/248599
ER -

References

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  6. Figiel T., Sucheston L., An application of Ramsey sets in analysis, Adv. Math. 20 (1976), 103-105. (1976) Zbl0325.46029MR0417757
  7. James R.C., Weakly compact sets, Trans. Amer. Math. Soc. 113 (1964), 129-140. (1964) Zbl0129.07901MR0165344
  8. Lévy P., Lecons d'analyse fonctionnelle, Gauthier-Villars, Paris, 1922. Zbl0043.32302
  9. Lévy P., Problèmes concrets d'analyse fonctionnelle, Gauthier-Villars, Paris, 1951. Zbl0155.18201MR0041346
  10. Lévy P., Quelques aspects de la pensée d'un mathématicien, Blanchard, Paris, 1970. Zbl0219.01020MR0268008
  11. Rudin W., Functional Analysis, McGraw-Hill, New York, 1973. Zbl0867.46001MR0365062

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