1
$\begingroup$

This question was asked at MSe before but with no answer.

I am struggling with the very last estimate in the proof of James' $\ell_1$-theorem. (Please see below an excerpt from Albiac and Kalton's fantastic book Topics in Banach space theory.)

I don't see what kind of manipulation with indices $i,j$ is done in order to arrive at the final estimate.

Also,

where exactly we use the assumption that $(x_n)$ is equivalent to the canonical basis of $\ell_1$?

The relevant excerpt:

The relevant excerpt is here.

$\endgroup$
1
  • 1
    $\begingroup$ I think the assumption is needed in order to guarantee that $M_n$ exists. $\endgroup$
    – Yemon Choi
    Nov 4, 2014 at 12:47

1 Answer 1

1
$\begingroup$

Look at Giesy's proof, which gives more.

D. P. Giesy, On a convexity condition in normed linear spaces, Trans. Amer. Math. Soc.125 (1966), 114-146.

For every cardinal number (finite or infinite) $N$, if $(x_a)_{a < N}$ is a set of unit vectors that is $K$ equivalent to the unit vector basis of $\ell_1^{N}$, then there are disjointly supported (relative to the basis $(x_a)_{a< N}$ unit vectors $(y_b)_{b < \sqrt{N}}$ that are $\sqrt{K}$ equivalent to the unit vector basis of $\ell_1^{\sqrt{N}}$. The proof is just an exercise. Break the basis $(x_a)_{a < N}$ into $\sqrt{N}$ disjoint subsets each of cardinality $\sqrt{N}$. If no piece "works", use the condition to get a unit vector that witness the non-working--that set of vectors must "work".

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.