Let A and B be sets of real sequences. Let F(A, D) denote the set of all functions f : R → R that preserve A and B in the sense that (f(an))∈ B for all sequences (an)∈ A. We characterize F(A,B) when .4 and B are convergent series, absolutely convergent series, relatively convergent series and divergent series.
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For a map p of N into itself, consider the induced transformation [...] of series in a topological vector space. Then such properties of this transformation as sending convergent series to convergent series, or convergent series to bounded series, or bounded series to bounded series (and a few more) are mutually equivalent. Moreover, they are equivalent to an intrinsic property of p which reduces to those found by Agnew and Pleasants (in the case of permutations) and Witula (in the general case) as necessary and sufficient conditions for the above transformation to preserve convergence of scalar series. In the paper, the scalar case is treated first using simple Banach space methods, and then the result is easily extended to the general setting.
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