View source for Banach-Saks property

A [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110110/b1101101.png" /> is said to have the Banach–Saks property if bounded sequences admit subsequences having norm-convergent arithmetic means (such subsequences are called Banach–Saks sequences). The origin of the Banach–Saks property can be traced back to a result of S. Mazur: If a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110110/b1101102.png" /> in a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110110/b1101103.png" /> is weakly convergent to some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110110/b1101104.png" />, then there is a sequence formed by convex combinations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110110/b1101105.png" /> that converges in norm to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110110/b1101106.png" />. (The proof is just an application of the [[Hahn–Banach theorem|Hahn–Banach theorem]]: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110110/b1101107.png" /> belongs by hypothesis to the weak closure of the convex hull of the sequence, and the convex closed sets coincide for the weak and norm topologies.) It is then a natural question to ask if that convex combination can be chosen to be the arithmetic means. S. Banach and S. Saks [[#References|[a1]]] proved that this is the case in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110110/b1101108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110110/b1101109.png" />. S. Kakutani [[#References|[a8]]] extended this result to uniformly convex spaces. Later, T. Nishiura and D. Waterman [[#References|[a9]]] proved that a space with the Banach–Saks property must be reflexive (cf. [[Reflexive space|Reflexive space]]) and W. Schlenk [[#References|[a13]]] proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110110/b11011010.png" /> satisfies the weak Banach–Saks property: weakly null (instead of bounded) sequences have Banach–Saks subsequences. J. Schreier [[#References|[a12]]] showed that a certain weakly null sequence of indicator functions on the compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110110/b11011011.png" /> contains no Banach–Saks subsequences; thus, not all spaces have the weak Banach–Saks property. An example of A. Baernstein II [[#References|[a5]]] shows that not all reflexive spaces have the Banach–Saks property.

Subsequences of Banach–Saks sequences need not be Banach–Saks. For example, consider a sequence formed by an infinity of zeros and ones. The so-called Ramsey methods have had a great impact in the study of the Banach–Saks property and, in general, of properties depending on the extraction of subsequences, because they introduce some order into this phenomenon. The basic [[Ramsey theorem|Ramsey theorem]] asserts that given a sequence and a property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110110/b11011012.png" /> that pairs may have, either there is a subsequence for which all pairs have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110110/b11011013.png" /> or there is a subsequence for which no pair has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110110/b11011014.png" />. Some topological work allows one to replace, in some cases,  "pair"  by  "subsequence" . For instance, a bounded sequence in a Banach space either contains a subsequence such that all its further subsequences are Banach–Saks or it contains a subsequence having no Banach–Saks subsequences. Classical, as well as Ramsey, methods are described in full detail in [[#References|[a4]]]; a survey of counterexamples and new related properties is [[#References|[a3]]].

Regarding stability properties, J.R. Partington [[#References|[a10]]] proved that the Banach–Saks property is stable under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110110/b11011015.png" />-sums, and W. Schachermayer [[#References|[a11]]] and J. Bourgain (see [[#References|[a7]]]) examples of Banach spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110110/b11011016.png" /> with the Banach–Saks property but for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110110/b11011017.png" /> does not have the Banach–Saks property. B. Beauzamy [[#References|[a6]]] studied the behaviour of the Banach–Saks property under interpolation. It does not pass from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110110/b11011018.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110110/b11011019.png" /> (or vice versa). It is unknown what the dual of a space with the Banach–Saks property looks like.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Banach,   S. Saks,   "Sur la convergence forte dans les champs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110110/b11011020.png" />"  ''Studia Math.'' , '''2'''  (1930)  pp. 51–57</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Beauzamy,   J.-T. Lapresté,   "Modèles étalés des espaces de Banach" , Hermann  (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.M.F. Castillo,   "Extraction of subsequences in Banach spaces"  ''Extracta Math.'' , '''7'''  (1992)  pp. 77–88</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Diestel,   "Sequences and series in Banach spaces" , ''GTM'' , '''92''' , Springer  (1984)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Baernstein II,   "On reflexivity and summability"  ''Studia Math.'' , '''42'''  (1972)  pp. 91–94</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  B. Beauzamy,   "Espaces d'interpolation réels: topologie et géométrie" , ''Lecture Notes in Mathematics'' , '''666''' , Springer  (1978)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  S. Guerre,   "La propriété de Banach–Saks ne pase pas de <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110110/b11011021.png" /> à <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110110/b11011022.png" />, d'áprés J. Bourgin"  ''Sem. Anal. Fonctionnelle Ecole Polytechn. Palaiseau''  (1979–1980)  pp. Exp. 8</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  S. Kakutani,   "Weak convergence in uniformly convex spaces"  ''Math. Inst. Osaka Imp. Univ.''  (1938)  pp. 165–167</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  T. Nishiura,   D. Waterman,   "Reflexivity and summability"  ''Studia Math.'' , '''23'''  (1963)  pp. 53–57</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  J.R. Partington,   "On the Banach–Saks property"  ''Math. Proc. Cambridge Philos. Soc.'' , '''82'''  (1977)  pp. 369–374</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  W. Schachermayer,   "The Banach–Saks property is not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110110/b11011023.png" />-hereditary"  ''Israel J. Math.'' , '''40'''  (1981)  pp. 340–344</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  J. Schreier,   "Ein Gegenbeispiel zur Theorie der swachen Konvergenz"  ''Studia Math.'' , '''2'''  (1930)  pp. 58–62</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  W. Schlenk,   "Sur les suites faiblement convergents dans l'espace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110110/b11011024.png" />"  ''Studia Math.'' , '''25'''  (1969)  pp. 337–341</TD></TR></table>