Banach-Saks property is a property of certain normed vector spaces stating that every bounded sequence of points in the space has a subsequence that is convergent in the mean (also known as Cesàro summation or limesable). Specifically, for every bounded sequence in the space, there exists a subsequence such that the sequence
is convergent (in the sense of the norm). Sequences satisfying this property are called Banach-Saks sequences.
The concept is named after Polish mathematicians Stefan Banach and Stanisław Saks, who extended Mazur's theorem, which states that the weak limit of a sequence in a Banach space is the limit in the norm of convex combinations of the sequence's terms. They showed that in Lp(0,1) spaces, for , there exists a sequence of convex combinations of the original sequence that is also Cesàro summable. This result was further generalized by Shizuo Kakutani to uniformly convex spaces. introduced the "weak Banach-Saks property", replacing the bounded sequence condition with a sequence weakly convergent to zero, and proved that the space has this property. The definitions of both Banach-Saks properties extend analogously to subsets of normed spaces.
Theorems and examples Every Banach space with the Banach-Saks property is reflexive. However, there exist reflexive spaces without this property, with the first example provided by Albert Baernstein. Julian Schreier provided the first example of a space (the so-called Schreier space) lacking the weak Banach-Saks property. He also proved that the space of continuous functions on the ordinal lacks this property. ℓp-sums of spaces with the Banach-Saks property retain this property. There exists a space with the Banach-Saks property for which the space (square-integrable functions in the Bochner sense with values in ) lacks this property. The image of a strictly additive vector measure has the Banach-Saks property. If a Banach space has a dual space that is uniformly convex, then has the Banach-Saks property. The dual space of the Schlumprecht space has the Banach-Saks property.
p-BS property and Banach-Saks index For a fixed real number , a bounded sequence in a Banach space is called a p-BS sequence if it contains a subsequence such that
A Banach space is said to have the p-BS property if every sequence weakly convergent to zero contains a subsequence that is a p-BS sequence. The p-BS property does not generalize the Banach-Saks property. Notably, every Banach space has the 1-BS property. The set
is of the form or , where . If , the Banach-Saks index of the space is defined as ; if , then . For example, the space has the 2-BS property.
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