Title: The projection methods in countably normed spaces
Abstract: It is well known that a normed space is uniformly convex (smooth) if and only if its dual space is uniformly smooth (convex). We extend the notions of uniform convexity (smoothness) from normed spaces to countably normed spaces ‘in which there is a countable number of compatible norms’. We get some fundamental links between Lindenstrauss duality formulas. A duality property between uniform convexity and uniform smoothness of countably normed spaces is also given. Moreover, based on the compatibility of those norms, it is interesting to show that ‘from any point in a real uniformly convex complete countably normed space, the nearest point to a nonempty convex closed subset of the space is the same for all norms’, which is helpful in further studies for fixed points.