Hahn-Banach theorems for $\kappa $-normed spaces

For a new class of topological vector spaces, namely $\kappa$-normed spaces, and associated quasisemilinear topological preordered space is defined and investigated. This structure arise naturally from the consideration of a $\kappa$-norm, that is a distance function between a point and a $G_{\delta}$-subset. For it, analogs of the Hahn-Banach theorem are proved.