Some counterexamples concerning strong M-bases of Banach spaces

M.S. Lambrou, William Longstaff

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    3 Citations (Scopus)

    Abstract

    A sequence of elements (f(n))(1)(infinity) of a real or complex Banach space X is an M-basis (of X) if (Vn=1Fn)-F-infinity=X and there exists a biorthogonal sequence of elements (f*(n))(1)(infinity) of X* satisfying boolean AND(n=1)(infinity)ker f*(n)=(0). An M-basis (f(n))(1)(infinity) is a strong M-basis if, additionally, x is an element of V{f(n):f(n)*(x) not equal 0}, for every element x is an element of X. Let X be a Banach space having a (Schauder) basis. We show that there exists a strong M-basis of X which is not finitely series summable. It follows that there is an atomic Boolean subspace lattice on X, with one-dimensional atoms, that fails to have the strong rank one density property. We show that there is always an atomic Boolean subspace lattice on X, with precisely four atoms, that also fails to have this density property. Also, if X = c(0) or c, an example is given of a strong M-basis (f(n))(1)(infinity) of X such that V(n=1)(infinity)f*(n)=X* but with (f*n)(1)(infinity) failing to be a strong M-basis of X*. This partially follows from a description that is given of a class of strong M-bases of c(0), c, l(p) (1 less than or equal to p <infinity). (C) 1994 Academic Press, Inc.
    Original languageEnglish
    Pages (from-to)243-259
    JournalJournal of Approximation Theory
    Volume79
    Issue number2
    DOIs
    Publication statusPublished - 1994

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