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Upper and lower estimates for Schauder frames and atomic decompositions

Volume 231 / 2015

Kevin Beanland, Daniel Freeman, Rui Liu Fundamenta Mathematicae 231 (2015), 161-188 MSC: Primary 46B20; Secondary 41A65. DOI: 10.4064/fm231-2-4


We prove that a Schauder frame for any separable Banach space is shrinking if and only if it has an associated space with a shrinking basis, and that a Schauder frame for any separable Banach space is shrinking and boundedly complete if and only if it has a reflexive associated space. To obtain these results, we prove that the upper and lower estimate theorems for finite-dimensional decompositions of Banach spaces can be extended and modified to Schauder frames. We show as well that if a separable infinite-dimensional Banach space has a Schauder frame, then it also has a Schauder frame which is not shrinking.


  • Kevin BeanlandWashington and Lee University
    204 W. Washington St.
    Lexington, VA 24450, U.S.A.
  • Daniel FreemanDepartment of Mathematics
    and Computer Science
    Saint Louis University
    St. Louis, MO 63103, U.S.A.
  • Rui LiuDepartment of Mathematics and LPMC
    Nankai University
    Tianjin 300071, P.R. China
    Department of Mathematics
    Texas A&M University
    College Station, TX 77843, U.S.A.

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