Banach-Saks property

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 ${\displaystyle (x_{n})_{n}}$ in the space, there exists a subsequence ${\displaystyle (x_{n_{k}})_{k}}$ such that the sequence

${\displaystyle \left({\frac {x_{n_{1}}+\ldots +x_{n_{k}}}{k}}\right)_{k=1}^{\infty }}$

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 L_p(0,1) spaces, for ${\displaystyle 1<p<\infty }$, 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. Wiesław Szlenk introduced the "weak Banach-Saks property", replacing the bounded sequence condition with a sequence weakly convergent to zero, and proved that the space ${\displaystyle L_{1}(0,1)}$ has this property. The definitions of both Banach-Saks properties extend analogously to subsets of normed spaces.