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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Perfect matrix methods


Authors: D. J. Fleming and P. G. Jessup
Journal: Proc. Amer. Math. Soc. 29 (1971), 319-324
MSC: Primary 40.31
DOI: https://doi.org/10.1090/S0002-9939-1971-0279484-X
MathSciNet review: 0279484
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Abstract: Let $ {e_i} = ({\delta _{ij}})_{j = 1}^\infty ,\Delta = ({e_i})_{i = 1}^\infty $ and let A be an infinite matrix which maps E into E where E is an FK-space with Schauder basis $ \Delta $. Necessary and sufficient conditions in terms of the matrix A are obtained for E to be dense in the summability space $ {E_A} = \{ x \vert {Ax \in E\} }$ and conditions are obtained to guarantee that $ {E_A}$ has Schauder basis $ \Delta $. Finally it is shown that if weak and strong sequential convergence coincide in E then in $ {E_A}$ the series $ \sum {_k{x_k}{e_k}} $ converges to x strongly if and only if it converges to x weakly.


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DOI: https://doi.org/10.1090/S0002-9939-1971-0279484-X
Keywords: Perfect matrix method, associative matrix method, property P, type M, type $ {M^ \ast }$
Article copyright: © Copyright 1971 American Mathematical Society