Schur property

The Schur property , named after Issai Schur , is a property from the mathematical theory of normalized spaces ; it is a close relationship between the norm topology and the weak convergence .

definition

A normalized space has the Schur property if every weakly-convergent sequence is also norm -convergent .

More precisely, this means: If a sequence is in normalized space and is so that it is weak, i.e. for every continuous , linear functional of space in the basic body, then it already follows . ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle x\in X}$ ${\displaystyle x_{n}\rightarrow x}$ ${\displaystyle f(x_{n})\rightarrow f(x)}$ ${\displaystyle f}$ ${\displaystyle \|x_{n}-x\|\rightarrow 0}$

comment

Conversely, since the weak convergence always follows from the norm convergence, the Schur property can also be formulated in such a way that the norm topology and the weak topology have the same convergent consequences. It does not follow from this that the topologies match, because the sequences are not sufficient to describe the weak topology. In fact, these topologies only agree if the space is finite dimensional.

Examples

• Every finite-dimensional, standardized space has the Schur property, because then the standard topology and weak topology match.
• Schur's theorem : The sequence space has the Schur property. This was proven by Issai Schur in 1920, hence this property is named Schur. ${\displaystyle \ell ^{1}}$
• The following rooms do not have the Schur property. If the sequence is 1 in the -th position and 0 in all other positions, then one shows in the weak topology, but because of does not apply in the standard topology. ${\displaystyle (\ell ^{p},\|\cdot \|_{p}),\,1<p<\infty }$ ${\displaystyle e_{n}=(0,\ldots ,0,1,0,\ldots )\in \ell ^{p}}$ ${\displaystyle n}$ ${\displaystyle e_{n}\rightarrow 0}$ ${\displaystyle \|e_{n}\|_{p}=1}$ ${\displaystyle e_{n}\rightarrow 0}$

properties

• If a normalized space is isomorphic to a normalized space with the Schur property, this also has the Schur property. This is because isomorphic normalized spaces have homeomorphic weak topologies.
• Sub-spaces of rooms with Schur property also have the Schur property.
• The Schur property is not carried over to quotient spaces , because every separable Banach space is known to be a quotient space of , as is the space that does not have the Schur property. ${\displaystyle \ell ^{1}}$ ${\displaystyle \ell ^{2}}$
• Infinite-dimensional spaces with the Schur property are not reflexive .
• Every standardized space with the Schur property has the Radon-Riesz property .
• Every Banach space with the Schur property has the Dunford-Pettis property .
• In a Banach space with Schur property, a subset is weakly compact if and only if it is norm-compact.
• Banach spaces with the Schur property are weakly sequence-complete, i.e. every weak Cauchy sequence has a weak limit.
• The following result shows how closely the Schur property is linked to space : If a Banach space with the Schur property, then every closed, infinitely dimensional subspace contains a subspace that is too isomorphic. ${\displaystyle \ell ^{1}}$ ${\displaystyle X}$ ${\displaystyle \ell ^{1}}$

Individual evidence

1. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , definition 2.5.25
2. ^ Joseph Diestel: Sequences and Series in Banach Spaces. 1984, ISBN 0-387-90859-5 . Chapter VII, page 85, Schur's theorem
3. ↑ J. Schur: About linear transformations in the theory of infinite series , J. Reine and applied mathematics (1920), Volume 151, pages 79–111
4. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , corollary 2.8.11
5. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , Theorem 2.3.7
6. ^ Albrecht Pietsch: History of Banach Spaces and Linear Operators , Birkhäuser-Verlag, Chapter 3.5 Weak sequential completeness and the Schur property
7. ^ TJ Morrison: Functional Analysis: An Introduction to Banach Space Theory , John Wiley & Sons (2001), ISBN 0-471-37214-5 , Corollary 5.12