Orthogonality in smooth countably normed spaces

We generalize the concepts of normalized duality mapping, J -orthogonality and Birkhoff orthogonality from normed spaces to smooth countably normed spaces. We give some basic properties of J -orthogonality in smooth countably normed spaces and show a relation between J -orthogonality and metric projection on smooth uniformly convex complete countably normed spaces. Moreover, we define the J -dual cone and J -orthogonal complement on a nonempty subset S of a smooth countably normed space and prove some basic results about the J -dual cone and the J -orthogonal complement of S .