Banach-Saks property

The Banach-Saks property , named after Stefan Banach and Stanisław Saks , is a mathematical property from the theory of Banach spaces . For a restricted sequence it ensures the existence of a subsequence which converges in the arithmetic mean.

Definition and motivation

A Banach space has the Banach-Saks property if every bounded sequence in X has a Cesàro -convergent subsequence , that is, if there is one with . ${\displaystyle X}$ ${\displaystyle (x_{m})_{m}}$ ${\displaystyle (x_{m_{n}})_{n}}$ ${\displaystyle x\in X}$ ${\displaystyle \left\|{\frac {1}{N}}\sum _{n=1}^{N}x_{m_{n}}-x\right\|\rightarrow 0}$

Many authors abbreviate the Banach-Saks property as BSP (Banach-Saks property).

According to a well-known theorem by Mazur , the limit value of a weakly convergent sequence can be approximated by convex combinations of the sequence members in the norm topology . The question arises whether this can even be achieved using the arithmetic mean , at least after switching to a partial sequence . If one has to go over to partial sequences anyway, one can try to consider limited sequences instead of weakly convergent sequences, because, at least in reflexive spaces, in which the unit sphere is known to be weakly compact and therefore even weak according to the Eberlein-Šmulian theorem - is sequence- compact , one can select weakly convergent partial sequences from limited sequences. These considerations then lead to the definition given above.

Examples

• Hilbert spaces have the Banach-Saks property.
• The L ^p ([0,1]) -spaces,, have the Banach-Saks property. ${\displaystyle 1<p<\infty }$
• S.Kakutani : Evenly convex spaces have the Banach-Saks property, the converse does not apply.
• According to a theorem by T. Nishiura and D. Waterman , Banach spaces with the Banach-Saks property are reflexive, the reverse is not true. One therefore has the following classification

Uniformly convex super-reflexive Banach-Saks property reflexive. ${\displaystyle \Rightarrow }$ ${\displaystyle \Rightarrow }$ ${\displaystyle \Rightarrow }$

• Non-reflexive spaces, such as the sequential spaces or , are therefore examples of Banach spaces without a Banach-Saks property. ${\displaystyle c_{0}}$ ${\displaystyle \ell ^{1}}$

Inheritance

• The Banach-Saks property is inherited by closed subspaces and quotient spaces .
• Conversely, if a Banach space with a closed subspace , so that and have the Banach-Saks property, then also has the Banach-Saks property. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Y}$ ${\displaystyle X/Y}$ ${\displaystyle X}$
• If a Banach space with the Banach-Saks property and is too isomorphic, then it also has the Banach-Saks property. This inheritance property does not apply to uniformly convex spaces, because uniform convexity is a property of the norm. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle Y}$
• The Banach-Saks property is not inherited by the dual space .

Related terms

The p -Banach-Saks property

A Banach space has the -Banach-Saks property if every bounded sequence in a subsequence contains, for which there is one and a constant are with all . (The constant can depend on the sequence under consideration, but not on .) ${\displaystyle p}$ ${\displaystyle (x_{m})_{m}}$ ${\displaystyle X}$ ${\displaystyle (x_{m_{n}})_{n}}$ ${\displaystyle x\in X}$ ${\displaystyle C>0}$ ${\displaystyle \left\|\sum _{n=1}^{N}(x_{m_{n}}-x)\right\|\leq C\cdot N^{\frac {1}{p}}}$ ${\displaystyle N\in \mathbb {N} }$ ${\displaystyle C>0}$ ${\displaystyle N}$

From the -Banach-Saks property,, follows the Banach-Saks property, because . ${\displaystyle p}$ ${\displaystyle p>1}$ ${\displaystyle \left\|{\frac {1}{N}}\sum _{n=1}^{N}x_{m_{n}}-x\right\|={\frac {1}{N}}\left\|\sum _{n=1}^{N}(x_{m_{n}}-x)\right\|\leq {\frac {1}{N}}\cdot C\cdot N^{\frac {1}{p}}=C\cdot N^{{\frac {1}{p}}-1}\rightarrow 0}$

In their 1930 work, Stefan Banach and Stanisław Saks essentially showed that the L ^p ([0,1]) spaces for what is known today as the p -Banach-Saks property. This is historically the starting point for studying the Banach-Saks property. ${\displaystyle 1<p<\infty }$

The alternating Banach-Saks quality

Since Banach spaces with the Banach-Saks property are reflexive, the question arises as to which property, conversely, a reflexive tree must have in order to have the Banach-Saks property. The presented here property comes into play: A Banach space has the alternating Banach-Saks property, if every bounded sequence in a subsequence has, so in the norm topology converges. MI Ostrowskii showed the following characterization: ${\displaystyle X}$ ${\displaystyle (x_{m})_{m}}$ ${\displaystyle X}$ ${\displaystyle (x_{m_{n}})_{n}}$ ${\displaystyle \sum _{n=1}^{\infty }(-1)^{n}x_{m_{n}}}$

• A Banach space has the Banach-Saks property if and only if it is reflexive and has the alternating Banach-Saks property.

The weak Banach-Saks quality

A Banach space has the weak Banach-Saks property if every weak null sequence in X has a Cesàro-convergent subsequence , that is, there is an with . ${\displaystyle (x_{m})_{m}}$ ${\displaystyle (x_{m_{n}})_{n}}$ ${\displaystyle x\in X}$ ${\displaystyle \left\|{\frac {1}{N}}\sum _{n=1}^{N}x_{m_{n}}-x\right\|\rightarrow 0}$

Since weak null sequences are bounded, the weak Banach-Saks property follows from the Banach-Saks property. The spaces (proven by W. Schlenk ) and the sequence spaces and have the weak Banach-Saks property, but not the Banach-Saks property due to a lack of reflexivity. The function space and the sequence space are examples of Banach spaces without the weak Banach-Saks property. The weak Banach-Saks property is inherited on closed subspaces, but not on quotient spaces. ${\displaystyle L^{1}([0,1])}$ ${\displaystyle c_{0}}$ ${\displaystyle \ell ^{1}}$ ${\displaystyle C([0,1])}$ ${\displaystyle \ell ^{\infty }}$

For the p -Banach-Saks property there is a weak variant: A Banach space has the weak -Banach-Saks property ${\displaystyle p}$ , if any weak zero sequence in a subsequence contains, for which there is one and a constant are with . ${\displaystyle (x_{m})_{m}}$ ${\displaystyle X}$ ${\displaystyle (x_{m_{n}})_{n}}$ ${\displaystyle x\in X}$ ${\displaystyle C>0}$ ${\displaystyle \left\|\sum _{n=1}^{N}(x_{m_{n}}-x)\right\|\leq C\cdot N^{\frac {1}{p}}}$

From the p -Banach-Saks property follows the weak p -Banach-Saks property, because weak zero sequences are limited, and from the weak p -Banach-Saks property follows the weak Banach-Saks property . ${\displaystyle 1<p<\infty }$

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• S. Banach and S. Saks: Sur la convergence forte dans les champs L ^p , Studia Mathematica, Volume 2, pages 51-57 (1930).
• Jesus M. Castillo, Manuel Gonzales: Three-space Problems in Banach Space Theory, Lecture Notes in Mathematics, Volume 1667 (1997), ISBN 978-3-540-63344-0
• Joseph Diestel: Sequences and Series in Banach Spaces. 1984, ISBN 0-387-90859-5 .
• S. Kakutani: Weak convergence in uniformly convex spaces , Math. Inst. Osaka Imp. Univ. (1938) pp. 165-167
• T. Nishiura, D. Waterman: Reflexivity and summability , Studia Mathematica, Volume 23 (1963) pp. 53-57
• N. Okada: On the Banach-Saks property , Proceedings Japan Academy, Volume 60, Series A (1984), pages 246-248
• W. Schlenk: Sur les suites faiblement convergents dans l'espace L , Studia Mathematica, Volume 25 (1969) pages 337-341