Property (u) of Pelczynski

The property (u) of Pelczynski is a property of Banach spaces considered in the mathematical branch of functional analysis . It goes back to the Polish mathematician Aleksander Pełczyński and relates weak Cauchy sequences and weakly unconditional Cauchy series .

definition

A Banach space has the property (u) if for every weak Cauchy sequence there is a weakly unconditional Cauchy series in such that ${\displaystyle X}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \textstyle \sum _{k}u_{k}}$ ${\displaystyle X}$

${\displaystyle x_{n}-\sum _{k=1}^{n}u_{k}\quad {\xrightarrow[{n\to \infty }]{}}\quad 0}$   in the weak topology.

Every weak Cauchy sequence, apart from a weak zero sequence, corresponds to the sequence of the partial sums of a weakly unconditional Cauchy series, or WUC series for short.

Examples

• Every Banach space with an unconditional shudder base has the property (u).
• Every weakly sequential Banach space has the property (u).
• Every sub-Banach space of a Banach space with property (u) also has property (u).
• The function space of continuous functions on the interval [0,1] and the sequence space of bounded sequences do not have the property (u). The James space does not have the property (u) either. ${\displaystyle C([0,1])}$ ${\displaystyle \ell ^{\infty }}$

meaning

It can sometimes be difficult to prove that a Banach space does not have an unconditional basis. The absence of the property (u) results in an even stronger statement. A space without a property (u) is not even isomorphic to a sub-Banach space of a Banach space with unconditional basis because of the inheritance of this property on subspaces. According to the examples above, this applies to and the James room, in particular these rooms have no unconditional basis. ${\displaystyle C([0,1])}$

Also the space L ¹ ([0,1]) cannot be embedded in a Banach space with an unconditional basis. The proof of this theorem is more complex, because since L ¹ ([0,1]) has the property (u) as a weakly sequence-complete space, the above argumentation cannot be used.

The implications appear to be:

${\displaystyle X}$ has an unconditional basis.

${\displaystyle \Rightarrow \quad X}$ is isomorphic to a sub-Banach space of a Banach space with an unconditional basis.

${\displaystyle \Rightarrow \quad X}$ has the property (u).

The inversions do not apply. The spaces have an unconditional basis, but contain for subspaces without a basis. Hence the reverse of the first implication cannot hold. The above example L ¹ ([0,1]) shows that the second implication cannot be reversed either. ${\displaystyle \ell ^{p}}$ ${\displaystyle p\not =2}$

Individual evidence

1. ^ A. Pelczynski: A connection between weakly unconditional convergence and weakly completeness of Banach spaces , Bull. Acad. Polon. Sci. (1958), Volume 6, pp. 251-253
2. ^ CD Aliprantis, O. Burkinshaw: Positive operators , Springer-Verlag, ISBN 978-1-4020-5007-7 , definition 14.6
3. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , Definition 3.5.1
4. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , set 3.5.3
5. ^ CD Aliprantis, O. Burkinshaw: Positive operators , Springer-Verlag, ISBN 978-1-4020-5007-7 , Theorem 14.7
6. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , set 3.5.2
7. ^ CD Aliprantis, O. Burkinshaw: Positive operators , Springer-Verlag, ISBN 978-1-4020-5007-7 , example 14.8
8. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , set 3.5.4
9. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , Theorem 6.3.3
10. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , Theorem 5.2.10
11. ↑ AM Davie: The approximation problem for Banach spaces , Bull. London Math. Soc. (1973) Vol. 5, pp. 261-266
12. ↑ A. Szankowski: Subspaces without the approximation property , Israel J. Math. (1978), Volume 30, pages 123-129