Hahn-Banach Theorem and Duality Theory on non-Archimedean Locally Convex Spaces

Let k be a local field with valuation ring O and residue field $\overline{k}$. We extend Hahn-Banach theorem for the class of seminormed k-vector spaces to several classes of locally convex spaces and subspaces over k, O, and $\overline{k}$. We establish continuous duality theory for several classes of locally convex spaces over k, O, and $\overline{k}$.