Vol. 46, No. 2, 1973

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Dual spaces of certain vector sequence spaces

Ronald C. Rosier

Vol. 46 (1973), No. 2, 487–501

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 Verallgemeinerte Volkommene 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 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) is difficult to represent, but the dual of Λ{E} is shown to be easily representable for general Λ and E. For many special cases of Λ and E the dual of Λ{E} is of the form Λx{E′} where Λx is the α-dual of Λ and Eis the strong dual of E.

Mathematical Subject Classification 2000
Primary: 46A45
Received: 18 April 1972
Published: 1 June 1973
Ronald C. Rosier