Duality of <Emphasis Type="Italic">κ</Emphasis>-normed topological vector spaces and their applications

In this paper, the duality of κ-normed topological vector spaces X is defined and investigated, where X is over the field K = R, or K = C, or a non-Archimedean field. For such spaces, an analog of the Mackey-Arens theorem is proved. The conditional κ-normability of spaces L(X) of linear topological homeomorphisms of a locally convex κ-normed space X is studied, where the image of elements under the corresponding operations is in L(X). Cases where the κ-normability of a topological vector space implies its local convexity are investigated. Applications of κ-normed spaces for resolutions of differential equations and for approximations of functions in mathematical economics are given. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 52, Functional Analysis, 2008.