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
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.
