Complemented copies of $\ell^1$ and Pelczynski's property (V*) in Bochner function spaces

Details Complemented copies of $\ell^1$ and Pelczynski's property (V*) in Bochner function spaces Complemented copies of $\ell^1$ and Pelczynski's property (V*) in Bochner function spaces

1994-10-31

arxiv.org/abs/math/9410204v1

Mathematics

Functional Analysis

Scientific paper

Let $X$ be a Banach space and $(f_n)_n$ be a bounded sequence in $L^1(X)$. We prove a complemented version of the celebrated Talagrand's dichotomy i.e we show that if $(e_n)_n$ denotes the unit vector basis of $c_0$, there exists a sequence $g_n \in \text{conv}(f_n,f_{n+1},\dots)$ such that for almost every $\omega$, either the sequence $(g_n(\omega) \otimes e_n)$ is weakly Cauchy in $X \widehat{\otimes}_\pi c_0$ or it is equivalent to the unit vector basis of $\ell^1$. We then get a criterion for a bounded sequence to contain a subsequence equivalent to a complemented copy of $\ell^1$ in $L^1(X)$. As an application, we show that for a Banach space $X$, the space $L^1(X)$ has Pe\l czy\'nski's property $(V^*)$ if and only if $X$ does.

Affiliated with

Also associated with

No associations

Rating

Complemented copies of $\ell^1$ and Pelczynski's property (V*) in Bochner function spaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Complemented copies of $\ell^1$ and Pelczynski's property (V*) in Bochner function spaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Complemented copies of $\ell^1$ and Pelczynski's property (V*) in Bochner function spaces will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-716655