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

. 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} k ‾

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

\overline{k} k ‾

.