Subseries convergence in spaces with a Schauder basis

Swartz, Charles

doi:10.1090/S0002-9939-1995-1216826-9

Subseries convergence in spaces with a Schauder basis
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by Charles Swartz PDF

Proc. Amer. Math. Soc. 123 (1995), 455-457 Request permission

Abstract:

Let E be a Hausdorff topological vector space having a Schauder basis $\{ {b_i}\}$ and coordinate functionals $\{ {f_i}\}$. Let $\sigma (E,F)$ be the weak topology on E induced by $F = \{ {f_i}:i \in {\mathbf {N}}\}$. We show that if a series in E is subseries convergent with respect to $\sigma (E,F)$, then it is subseries convergent with respect to the original topology of E.

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