Banach-Saks property

Template:Short description 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 $(x_n{)}_n$ in the space, there exists a subsequence $(x_{n_k}{)}_k$ such that the sequence

${\left(\frac{x_{n_1}+\dots +x_{n_k}}{k}\right)}_{k=1}^{\mathrm{\infty }}$

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 $1<p<\mathrm{\infty }$, 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. Template:Interlanguage link introduced the "weak Banach-Saks property", replacing the bounded sequence condition with a sequence weakly convergent to zero, and proved that the space $L_1(0,1)$ 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 ${\omega }^{\omega }+1$ lacks this property.
• ℓ_p-sums of spaces with the Banach-Saks property retain this property.
• There exists a space $E$ with the Banach-Saks property for which the space $L_2(E)$ (square-integrable functions in the Bochner sense with values in $E$) lacks this property.
• The image of a strictly additive vector measure has the Banach-Saks property.
• If a Banach space $E$ has a dual space $E^{\ast }$ that is uniformly convex, then $E$ 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 $p⩾1$, a bounded sequence $(x_n{)}_n$ in a Banach space $X$ is called a p-BS sequence if it contains a subsequence $(x_{n_k}{)}_k$ such that

$\underset{m\in \mathbb{N}}{sup}\frac{1}{m^{\frac{1}{p}}}\Vert \sum _{i=1}^m{x}_{n_i}\Vert <\mathrm{\infty }.$

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

$\mathrm{\Gamma }(X)=\{p⩾1:X\text{ has the }p\text{-BS property}\}$

is of the form $[0,{\gamma }_0)$ or $[0,{\gamma }_0]$, where ${\gamma }_0⩾1$. If $\mathrm{\Gamma }(X)=[0,{\gamma }_0]$, the Banach-Saks index $\gamma (X)$ of the space $X$ is defined as $\gamma (X)={\gamma }_0$; if $\mathrm{\Gamma }(X)=[0,{\gamma }_0)$, then ${\gamma }_0=0$. For example, the space $L_2(0,1)$ has the 2-BS property.