Vol. 37, No. 1, 1971

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On unconditionally converging series and biorthogonal systems in a Banach space

Gregory Frank Bachelis and Haskell Paul Rosenthal

Vol. 37 (1971), No. 1, 1–5

Our main result is as follows: Let B be a Banach space containing no subspace isomorphic (linearly homeomorphic) to l, and let {(bnn)} be a biorthogonal sequence in B such that (βn) is total. If x B then n=1βn(x)bn converges unconditionally to x if and only if for every sequence (an) of 0’s and l’s there exists y B with βn(y) = anβn(x) for all n. This theorem improves previous results of Kadec and Pelczynski.

Similar results are obtained in the context of biorthogonal decompositions of a Banach space into separable subspaces.

Mathematical Subject Classification 2000
Primary: 46B15
Received: 5 February 1970
Revised: 17 July 1970
Published: 1 April 1971
Gregory Frank Bachelis
Haskell Paul Rosenthal