Geodesic
Limit points of arithmetic means of sequences in Banach spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) no. 1, pp. 97-106 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We shall prove the following statements: Given a sequence $\{a_n\}_{n=1}^{\infty}$ in a Banach space $\bold X$ enjoying the weak Banach-Saks property, there is a subsequence (or a permutation) $\{b_n\}_{n=1}^{\infty}$ of the sequence $\{a_n\}_{n=1}^{\infty}$ such that $$ \lim_{n\to\infty} {1\over n}\sum_{j=1}^n b_j=a $$ whenever $a$ belongs to the closed convex hull of the set of weak limit points of $\{a_n\}_{n=1}^{\infty}$. In case $\bold X$ has the Banach-Saks property and $\{a_n\}_{n=1}^{\infty}$ 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.
We shall prove the following statements: Given a sequence $\{a_n\}_{n=1}^{\infty}$ in a Banach space $\bold X$ enjoying the weak Banach-Saks property, there is a subsequence (or a permutation) $\{b_n\}_{n=1}^{\infty}$ of the sequence $\{a_n\}_{n=1}^{\infty}$ such that $$ \lim_{n\to\infty} {1\over n}\sum_{j=1}^n b_j=a $$ whenever $a$ belongs to the closed convex hull of the set of weak limit points of $\{a_n\}_{n=1}^{\infty}$. In case $\bold X$ has the Banach-Saks property and $\{a_n\}_{n=1}^{\infty}$ 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.
Classification : 40G05, 40H05, 46B20, 47F05
Keywords: Banach-Saks property; arithmetic means; limit points; subsequences; permutations of sequences 
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     author = {L\'avi\v{c}ka, Roman},
     title = {Limit points of arithmetic means of sequences in {Banach} spaces},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {97--106},
     year = {2000},
     volume = {41},
     number = {1},
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     zbl = {1040.46013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2000_41_1_a7/}
}
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Lávička, Roman. Limit points of arithmetic means of sequences in Banach spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) no. 1, pp. 97-106. http://geodesic.mathdoc.fr/item/CMUC_2000_41_1_a7/