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Schauder bases having many good block basic sequences

Volume 254 / 2020

Cory A. Krause Studia Mathematica 254 (2020), 199-218 MSC: Primary 46B03, 46B06, 46B25, 46B45; Secondary 05D10. DOI: 10.4064/sm190727-14-1 Published online: 17 April 2020

Abstract

In the study of asymptotic geometry in Banach spaces, a basic sequence which gives rise to a spreading model has been called a good sequence. It is well known that every normalized basic sequence in a Banach space has a subsequence which is good. We investigate the assumption that every normalized block tree relative to a basis has a branch which is good. This combinatorial property turns out to be very strong and is equivalent to the space being $1$-asymptotic $\ell _p$ for some $1\leq p\leq \infty $. We also investigate the even stronger assumption that every block basic sequence of a basis is good. Finally, using the Hindman–Milliken–Taylor theorem, we prove a stabilization theorem which produces a basic sequence all of whose normalized constant coefficient block basic sequences are good, and we present an application of this stabilization.

Authors

  • Cory A. KrauseLeTourneau University
    2100 S Mobberly
    Longview, TX 75602, U.S.A.
    e-mail

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