  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.
