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An Abstract Concept of the Sum of a Numerical Series

Published online by Cambridge University Press:  20 November 2018

William H. Ruckle
William H. Ruckle*
Affiliation:
Clemson University, Clemson, South Carolina
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Our aim in this paper, generally stated, is to formulate an abstract concept of the notion of the sum of a numerical series. More particularly, it is a study of the type of sequence space called “sum space”. The idea of sum space arose in connection with two distinct problems.

1.1 The Köthe-Toeplitz dual of a sequence space T consists of all sequences t such that st ∈ l1 (absolutely summable sequences) for each s∈T. It is known that if cs or bs is used in place of l1, an analogous theory of duality for sequence spaces can be developed (cf. [2]). What other spaces of sequences can play a rôle analogous to l1? This problem is treated in [6].

1.2. Let {xn, fn} be a complete biorthogonal sequence in (X, X*), where X is a locally convex linear topological space and X* is its topological dual space.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 4 , 01 August 1970 , pp. 863 - 874
Copyright
Copyright © Canadian Mathematical Society 1970

References

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