MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

By the work of E. Odell and Th. Schlumprecht, we know that the separable Hilbert space $\ell_2$ is arbitrarily distortable. But I don't know if an "asymptotic" version of their result is true. To state my question, let me introduce the following terminology:

Let $(X, \| \cdot \|)$ be a separable Banach space with a normalized Schauder basis $(e_n)$ and $C \geq 1$. Let us say that $X$ is asymptotically non-distortable with constant $C$ (and with respect to the basis $(e_n)$ of $X$) if for every equivalent norm $| \cdot |$ on $X$ there exists a semi-normalized block sequence $(v_n)$ of $(e_n)$ such that for every $k$, every $k \leq n_1 < ... < n_k$ and every pair $x$ and $y$ of vectors in the span of $\{ v_{n_1}, ..., v_{n_k} \}$ with $\|x\| = \|y\| = 1$ we have $|x| / |y| \leq C$.

Question: does there exist $C\geq 1$ such that the separable Hilbert space $\ell_2$ is asymptotically non-distortable with constant $C$ and with respect to its standard unit vector basis? IF this is true, then can we take $C$ to be $1+\epsilon$ for every $\epsilon > 0$?

Of course, a similar question can be asked for a general Banach space with a Schauder basis. I think that every asymptotic $\ell_1$ space is asymptotically non-distortable for some $C\geq 1$, and for Tsirelson's space we can take $C$ to be $2+\epsilon$ for every $\epsilon > 0$. Let me recall that there exist arbitrarily distortable asymptotic $\ell_1$ spaces.

share|cite|improve this question

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.