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 r(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 r(k). We establish continuous duality theory for several classes of locally convex spaces over k, O, and r(k).