On topological sequence spaces

We denote by ω the linear space of all sequences of real or complex numbers. A linear subspace of ω is called a sequence space. A subset A of ω is solid if whenever x ∈ A and |yi| ≤ |xi| for each i, then y ∈ A. The theory of solid sequence spaces, topologized in a variety of ways, has been developed in considerable detail, in particular by Köthe and Toeplitz (13) and subsequently by Köthe (see, for example (12)). These results have been generalized to function spaces by Dieudonné(6), to vector-valued sequence spaces by Pietsch (18), to vector spaces with a Boolean algebra of projections by Cooper ((4), (5)), and in the real case, to partially-ordered spaces by Luxemburg and Zaanen (see, for example (14)) and Fremlin (8). This last generalization shows that many of the properties of solid sequence spaces depend upon their order structure, rather than upon their structure as sequence spaces.