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.^[1] This result was further generalized by Shizuo Kakutani to uniformly convex spaces.^[2] 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.^[3] The definitions of both Banach-Saks properties extend analogously to subsets of normed spaces.

• Every Banach space with the Banach-Saks property is reflexive.^[4] However, there exist reflexive spaces without this property, with the first example provided by Albert Baernstein.^[5]
• 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 ${\displaystyle \omega ^{\omega }+1}$ lacks this property.^[6]
• ℓ_p-sums of spaces with the Banach-Saks property retain this property.^[7]
• There exists a space ${\displaystyle E}$ with the Banach-Saks property for which the space ${\displaystyle L_{2}(E)}$ (square-integrable functions in the Bochner sense with values in ${\displaystyle E}$) lacks this property.^[8]
• The image of a strictly additive vector measure has the Banach-Saks property.^[9][10]
• If a Banach space ${\displaystyle E}$ has a dual space ${\displaystyle E^{*}}$ that is uniformly convex, then ${\displaystyle E}$ has the Banach-Saks property.^[11]
• The dual space of the Schlumprecht space has the Banach-Saks property.^[12]

For a fixed real number ${\displaystyle p\geqslant 1}$, a bounded sequence ${\displaystyle (x_{n})_{n}}$ in a Banach space ${\displaystyle X}$ is called a p-BS sequence if it contains a subsequence ${\displaystyle (x_{n_{k}})_{k}}$ such that

${\displaystyle \sup _{m\in \mathbb {N} }{\frac {1}{m^{\frac {1}{p}}}}{\Bigg \|}\sum _{i=1}^{m}x_{n_{i}}{\Bigg \|}<\infty .}$

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.^[13][14] The p-BS property does not generalize the Banach-Saks property. Notably, every Banach space has the 1-BS property. The set

${\displaystyle \Gamma (X)=\{p\geqslant 1\colon \,X{\text{ has the }}p{\text{-BS property}}\}}$

is of the form ${\displaystyle [0,\gamma _{0})}$ or ${\displaystyle [0,\gamma _{0}]}$, where ${\displaystyle \gamma _{0}\geqslant 1}$. If ${\displaystyle \Gamma (X)=[0,\gamma _{0}]}$, the Banach-Saks index ${\displaystyle \gamma (X)}$ of the space ${\displaystyle X}$ is defined as ${\displaystyle \gamma (X)=\gamma _{0}}$; if ${\displaystyle \Gamma (X)=[0,\gamma _{0})}$, then ${\displaystyle \gamma _{0}=0}$. For example, the space ${\displaystyle L_{2}(0,1)}$ has the 2-BS property.^[13][14]