Bessaga-Pelczynski selection principle

The Bessaga-Pelczynski selection principle is a proposition from the mathematical subfield of functional analysis , which ensures the existence of basic sequences in any infinite-dimensional Banach spaces . It is named after the Polish mathematicians Czesław Bessaga and Aleksander Pełczyński , who published it in 1958.

First formulation of the sentence

In a Banach space, let it be a shudder basis with base constant and coefficient functionals . Next is a sequence in with ${\displaystyle X}$ ${\displaystyle (e_{n})_{n\in \mathbb {N} }}$ ${\displaystyle K}$ ${\displaystyle (e_{n}^{*})_{n\in \mathbb {N} }}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle X}$

• ${\displaystyle \textstyle \inf _{n\in \mathbb {N} }\|x_{n}\|>0}$
• ${\displaystyle \textstyle \lim _{n\to \infty }e_{m}^{*}(x_{n})=0}$   for everyone   . ${\displaystyle m\in \mathbb {N} }$

Then contains a subsequence that is congruent to a block basis sequence of . In addition, one can choose the partial sequence for each so that its base constant is at most . ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (x_{n_{k}})_{k\in \mathbb {N} }}$ ${\displaystyle (y_{k})_{k\in \mathbb {N} }}$ ${\displaystyle (e_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle K+\varepsilon }$

Second formulation of the sentence

Let it be a weak zero sequence with for all in an infinitely dimensional Banach space. Then contains a subsequence that is a base sequence. ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \|x_{n}\|=1}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$

In this formulation is called "Bessaga-Pelczynski Selection Principle (Utility Grade)", there are applications of this formulation of the selection principle.

Relationship between the formulations

The first formulation is stronger, we show here how the second can be derived from the first: It is sufficient to consider the sub-annex space generated by . This is generated in a countable manner and is therefore a separable space . As such, it can be understood as a subspace of the function space according to the Banach-Mazur theorem , and this has a shudder basis . Now the first formulation can be used, because the two conditions on the sequence result from and in the weak topology. In particular, contains a subsequence that is a base sequence. ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle C([0,1])}$ ${\displaystyle (e_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \|x_{n}\|=1}$ ${\displaystyle x_{n}\rightarrow 0}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$

Application: existence of basic sequences

• Every infinite-dimensional Banach space contains a basis sequence.

Proof: It is sufficient to consider infinitely dimensional, separable Banach spaces , and according to the Banach-Mazur theorem these can be regarded as subspace of without restriction . Let it be a shudder basis of with coefficient sequence . Since is infinitely dimensional, one can easily construct a sequence in with and for all . Then fulfills the requirements of the first formulation with regard to the Banach space and we get a subsequence that is the base sequence, and this is even in . ${\displaystyle X}$ ${\displaystyle C([0,1])}$ ${\displaystyle (e_{n})_{n\in \mathbb {N} }}$ ${\displaystyle C([0,1])}$ ${\displaystyle (e_{n}^{*})_{n\in \mathbb {N} }}$ ${\displaystyle X}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle X}$ ${\displaystyle \|x_{n}\|=1}$ ${\displaystyle e_{k}^{*}(x_{n})=0}$ ${\displaystyle k=1,\ldots ,n}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle C([0,1])}$ ${\displaystyle X}$

Individual evidence

1. ↑ C. Bessaga, A. Pelczynski: On bases and unconditional convergence of series in Banach spaces , Studia Mathematica (1958), Volume 17, Part 2, pages 151-164, Theorem 3
2. ^ Joseph Diestel: Sequences and series in Banach spaces , Springer-Verlag (1984), ISBN 0-387-90859-5 , Chapter V: Basic Sequences , page 46
3. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , set 1.3.10
4. ^ Joseph Diestel: Sequences and series in Banach spaces , Springer-Verlag (1984), ISBN 0-387-90859-5 , Chapter V: Basic Sequences , page 42
5. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , Theorem 1.4.4