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Schauder bases and decompositions in locally convex spaces

Published online by Cambridge University Press:  24 October 2008

J. H. Webb
J. H. Webb
Affiliation:
University of Cape Town, Rondesbosch C.P., South Africa
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Extract

Let E[τ] be a locally convex Hausdorif topological vector space, with a Schauder basis {xi, x′j where

for each x ∈ E. The partial summation operator Sn, defined by

is a linear operator on E, whose definition extends at once to a linear operator mapping (E′)* into E, where (E′)* is the algebraic dual of E′. The dual of Sn is the operator S′n, mapping E* into E′, defined by

and 〈Snx, x′〉 = 〈x, S′nx′〉 for each x ∈ (E′)*. It is easy to see that S′nx′ → x′ with respect to the weak topology σ(E′, E) for each x′ ∈ E′.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 76 , Issue 1 , July 1974 , pp. 145 - 152
Copyright
Copyright © Cambridge Philosophical Society 1974

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References

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