Perfect matrix methods

Fleming, D. J.; Jessup, P. G.

doi:10.1090/S0002-9939-1971-0279484-X

Perfect matrix methods
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by D. J. Fleming and P. G. Jessup

Proc. Amer. Math. Soc. 29 (1971), 319-324

DOI: https://doi.org/10.1090/S0002-9939-1971-0279484-X

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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 ${E_A} = \{ x | {Ax \in E\} }$ 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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