This article is an investigation
of certain spaces of sequences with values in a locally convex space analogous to the
generalized sequence spaces introduced by Pietsch in his monograph VerallgemeinerteVolkommene Folgenräume (Akademie-Verlag, Berlin, 1962). Pietsch combines a
perfect sequence space Λ and a locally convex space E to obtain the space Λ(E) of all
E valued sequences x = (xn) such that the scalar sequence (⟨a,xn⟩) is in Λ for every
a ∈ E′. Define Λ{E} to be the space of all E valued sequences x = (xn)
such that the scalar sequence (p(xn)) is in Λ for every continuous seminorm
p on E. The spaces Λ(E) and Λ{E} are topologized using the topology
of E and a certain collection ℳ of bounded subsets of Λx, the α-dual of
Λ.
The criteria for bounded sets, compact sets, and completeness are similar for both
spaces. The significant difference lies in the duality theory. The dual of Λ(E)ℳ is
difficult to represent, but the dual of Λ{E}ℳ is shown to be easily representable for
general Λ and E. For many special cases of Λ and E the dual of Λ{E}ℳ is of
the form Λx{E′} where Λx is the α-dual of Λ and E′ is the strong dual of
E.
