Let be your countabl dense subset of , and the sequence obtained by applying the Gram-Schmidt process to . Then let and be two elements in and then applying G-S to the first vectors we conclude that . So is an orthonormal set.
Now, without loss of generality, we may assume that is linearly independent, and since is contained in for every we have . Let , since is linearly independent (actually orthogonal) and we have (remember that all basis for a vector space have the same cardinality) that is a basis for and so . And you're finished