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• W2171719488 abstract "A complete biorthogonal sequence in a topological vector space X is a double sequence (xi, fi) such that (1) each xi ∈ X and each fi ∈ X* the dual space of X ; (2) the closed linear span of {xi} is dense in X, (3) fi (xj) = 1 if i = j and is 0 otherwise, and (4) fi (z) = 0 for each i only when z = 0. If x ∈ X and f ∈ X* the numerical series $${{sumlimits_{i} f }_{i}}(x)f({{x}_{i}}) $$ will converge to f(x) if (a) x is a finite linear combination of (xi); (b) f is a.finite linear combination of (fi) or (c) (xi) is a Schauder basis of X. In other cases (1) may not converge, or z if it does it may not converge to f (x); but there may be some method of series summability that will associate the “correct” sum, i. e. f (x), to the sequence (fi(x) f (xi)). In order to determine conditions under which there is such a method of summability we study four sequence spaces associated with the biorthogonal sequence (xi, fi), namely S = {(fi (x)): x ∈ X}, that represents the space X,Sf= { (f(xi)) : f ∈ X*}, that represents the dual space X* of X, S(S), the series space of S that consists of the linear span of all sequences of the form st where s ∈ S and t ∈ Sf, the multiplier space M (S) consisting of all sequences u such that US ∈ S whenever s ∈ S. We will discuss how conditions on these four spaces result in summability properties of the biorthogonal sequence (xi, fi) and also on the topology of the space X." @default.
• W2171719488 created "2016-06-24" @default.
• W2171719488 creator A5014139778 @default.
• W2171719488 date "1999-01-01" @default.
• W2171719488 modified "2023-09-26" @default.
• W2171719488 title "Series summability of complete biorthogonal sequences" @default.
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• W2171719488 doi "https://doi.org/10.1007/978-1-4612-2236-1_3" @default.
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