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The uniform forward sets of \(C(X)\) and uniform convergence of operator series. (English) Zbl 1145.46004

Summary: We introduce the uniform forward sets of \(C(X)\), the space of \(X\)-valued convergent sequences, and show that each totally bounded set of \(C(X)\) is a uniform forward set, moreover, we prove that the uniform forward sets of \(C(X)\) are just the largest subset family of \(C(X)\) on which each \(C(X)\)-evaluation convergent operator series is uniformly convergent.

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)
46B99 Normed linear spaces and Banach spaces; Banach lattices
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