It is well known that a normed space Eis a uniformly convex (smooth) normed space if and only if its dual  is uniformly smooth (convex). We extend these geometric properties to seminormed spaces and then introduce definitions of uniformly convex (smooth) countably seminormed spaces. A new vision of the completion of countably seminormed space was helpful in our task. We get some fundamental links between Lindenstrauss duality formulas. A duality property between uniform convexity and uniform smoothness of countably seminormed space is also given.

countably seminormed space, Fréchet spaces, uniformly convex (smooth) normed space, completion of countably seminormed space, uniformly convex (smooth) countably seminormed space, the dual of a countably seminormed space.