Base sequence

Basic sequences are used in the mathematical sub-area of functional analysis to investigate Banach spaces. These are sequences that are a shudder base in the subspace they create . Not every separable Banach space has a shudder basis, but there are always basis sequences.

definition

A sequence in a Banach space is called a basis sequence if a shudder basis is in, that is, in the closed , linear envelope of the elements . The basic sequence is called normalized if all have norm 1. ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle X}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle [(x_{n})_{n\in \mathbb {N} }]}$ ${\displaystyle x_{n}}$ ${\displaystyle x_{n}}$

Two basic consequences and in Banach spaces or hot equivalent if for every sequence the series of scalars iff in converged when in converged. This is exactly the case if there is a Banach space isomorphism with for all . ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (y_{n})_{n\in \mathbb {N} }}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle (\alpha _{n})_{n\in \mathbb {N} }}$ ${\displaystyle \textstyle \sum _{n=1}^{\infty }\alpha _{n}x_{n}}$ ${\displaystyle X}$ ${\displaystyle \textstyle \sum _{n=1}^{\infty }\alpha _{n}y_{n}}$ ${\displaystyle Y}$ ${\displaystyle T:[(x_{n})_{n\in \mathbb {N} }]\rightarrow [(y_{n})_{n\in \mathbb {N} }]}$ ${\displaystyle Tx_{n}=y_{n}}$ ${\displaystyle n\in \mathbb {N} }$

Two basis sequences and in Banach spaces or are called congruent if there is a Banach space isomorphism with for all . Obviously, congruent basis sequences are equivalent, the inversion of this statement fails because a Banach space isomorphism in general cannot be continued from one to another. ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (y_{n})_{n\in \mathbb {N} }}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle T:X\rightarrow Y}$ ${\displaystyle Tx_{n}=y_{n}}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle T:[(x_{n})_{n\in \mathbb {N} }]\rightarrow [(y_{n})_{n\in \mathbb {N} }]}$ ${\displaystyle X}$ ${\displaystyle Y}$

Examples

• Every shudder basis in a Banach space is a basis sequence, for example the canonical bases in the sequence spaces or the Haar system in the spaces L ^p ([0,1]) , where . ${\displaystyle \ell ^{p}}$ ${\displaystyle 1\leq p<\infty }$
• The sequence of the Rademacher functions in every space L ^p ([0,1]) ,, is a basis sequence which is equivalent to the canonical basis in , as can easily be seen from the Chinchin inequality . ${\displaystyle 1\leq p<\infty }$ ${\displaystyle \ell ^{2}}$

The Grinblum Criterion

The Grinblum criterion, named after the Russian mathematician Maximilian Michailowitsch Grinblum , decides whether a sequence in a Banach space is a basic sequence. Accordingly, a basic sequence is in if and only if all are different from 0 and there is a constant with ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle X}$ ${\displaystyle x_{n}}$ ${\displaystyle K>0}$

${\displaystyle \left\|\sum _{k=1}^{m}\alpha _{k}x_{k}\right\|\,\leq \,K\,\left\|\sum _{k=1}^{n}\alpha _{k}x_{k}\right\|}$

for every sequence of scalars and natural numbers with . ${\displaystyle (\alpha _{k})_{k\in \mathbb {N} }}$ ${\displaystyle m,n}$ ${\displaystyle m\leq n}$

The smallest , which satisfies the above inequality for all scalars and , is called the base constant of the base sequence. This is nothing more than the base constant of the shudder base in Banach space . ${\displaystyle K}$ ${\displaystyle \alpha _{k}}$ ${\displaystyle m<n}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle [(x_{n})_{n\in \mathbb {N} }]}$

Existence of basis sequences

The Bessaga-Pelczynski selection principle says that in an infinite-dimensional Banach space every sequence with for all that converges weakly to 0 contains a subsequence, which is the base sequence. From this it follows in particular ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \|x_{n}\|=1}$ ${\displaystyle n\in \mathbb {N} }$

• Every infinite-dimensional Banach space contains a closed subspace that has a shudder base.

The question immediately arises whether every infinitely dimensional Banach space even has a closed subspace with an unconditional shudder base . This problem remained open for a long time until William Timothy Gowers and Bernard Maurey set a negative example in 1993.

Basic block sequences

Let it be the shudder basis of a Banach space . A basic block sequence of is a sequence of vectors with ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle X}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (u_{n})_{n\in \mathbb {N} }}$ ${\displaystyle u_{n}\not =0}$

${\displaystyle u_{n}=\sum _{k=p_{n-1}+1}^{p_{n}}\alpha _{k}x_{k}}$   whereby . ${\displaystyle 0=p_{0}<p_{1}<p_{2}<\ldots ,\quad \alpha _{k}\in \mathbb {R} }$

They arise from the formation of blocks, which is where the name block base sequence comes from. In addition, one can show that there is actually a basic sequence whose constant is not greater than the basic constant of . ${\displaystyle u_{n}}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (u_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$

This is an important method of gaining new ones from given basic sequences by building blocks. It can be shown that the normalized block basis sequences formed from the canonical bases of or are equivalent to the starting basis. This leads to significant consequences for the structure of these sequence spaces. ${\displaystyle c_{0}}$ ${\displaystyle \ell ^{p}}$

Individual evidence

1. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , Definitions 1.1.8 and 1.3.1
2. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , Definition 1.3.8
3. ↑ MM Grinblum: Некоторые теоремы о базисе в пространстве типа (B) (Some sentences about bases in rooms of type (B)) , CR Docl. Akad. Sci. URSS (1941) Vol. 31, pp. 428-432
4. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , set 1.1.9
5. ^ Joseph Diestel: Sequences and series in Banach spaces , Springer-Verlag (1984), ISBN 0-387-90859-5 , Chapter V: Basic Sequences
6. ^ WT Gowers, B. Maurey: The unconditional basic sequence problem , J. Amer. Math. Soc. (1993), Vol. 6, pp. 851-874
7. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , Definition 1.3.4 and Lemma 1.3.5
8. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , Chapter 2: The Classical Sequence Spaces