Duality of $\kappa$-normed topological vector spaces and their applications

A duality of $\kappa$-normed topological vector spaces is defined and investigated. For such spaces the analog of the Mackey-Arens theorem is proved. There are investigated cases, when $\kappa$-normability of a topological vector space implies its local convexity. There are given applications of $\kappa$-normed spaces for resolutions of differential equations and for approximations of functions in mathematical economy.