Vol. 42, No. 1, 1972

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Topologies on sequences spaces

William Henry Ruckle

Vol. 42 (1972), No. 1, 235–249

A study is made of two means to topologize a space of sequences. The first method rests upon the duality of every sequence space S with the sequence space φ (finitely 0) by means of the form

((aj),(bj)) =    ajbj(aj) ∈ S,(bj) ∈ φ.

The second method is a generalization of the Köthe-Toeplitz duality theory. The Köthe dual Sα of a sequence space S consists of all (bj) such that (ajbj) l1 (absolutely convergent series) for (aj) S. Other spaces may take the role of l1 in the above definition. A means to construct a topology on S is determined using this generalized dual. Finally, a particularly suitable type of space (the sum space) to play the role of l1 is defined.

Mathematical Subject Classification 2000
Primary: 46A45
Received: 3 March 1971
Revised: 6 June 1971
Published: 1 July 1972
William Henry Ruckle