Let $p\in [1,\infty)\setminus\{2\}$. Suppose $(e_n)$ is a basic sequence in $\ell_p$ (or $L_p$) equivalent to the basis of $\ell_p$ ($L_p$). Is there a subsequence $(e_{n_k})$ such that $[e_{n_k}]$ is complemented?

  • 1
    $\begingroup$ Yes, and it is simple fact that is proved in many books, including Albiac-Kalton. This is not an MO level question. Now if you had asked about $L_p$... $\endgroup$ – Bill Johnson Mar 29 '12 at 17:16
  • $\begingroup$ Thank you. Indeed, I am also interested in the $L_p$ case as well so let me modify my question. $\endgroup$ – Olaf Kummers Mar 29 '12 at 17:43

The answer is yes also for $L_p$, but I don't know a good book reference. For $2<p<\infty$, this is contained the paper of Kadec and Pelczynski--it is their second dichotomy theorem. Actually, they get that a normalized weakly null sequence has a subsequence that is either equivalent to an orthonormal sequence (in which case its closed span is automatically complemented) or has, for every $\epsilon > 0$, a subsequence that is $1+\epsilon$-equivalent to the unit vector basis for $\ell_p$ and spans a subspace that is $1+\epsilon$-complemented.

For $1\le p <2$, I think the result was pointed out by Pelczynski but I don't know a reference. It follows from arguments like those in Wojtaszczyk's book characterizing weak compactness in $L_1$. You can find an outline of the argument in a paper I wrote with G. Schechtman:

Multiplication operators on L(Lp) and lp-strictly singular operators, J. European Math. Society 10 1105-1119 (2008), which you can download from my home page.

EDIT July 7, 2012: The result above that I attributed to Pelczynski is actually due to Enflo and Rosenthal:

Enflo, Per; Rosenthal, Haskell P. Some results concerning Lp(μ)-spaces. J. Functional Analysis 14 (1973), 325–348.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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