Suppose that {e_i ; i Î I} is a Hamel basis for X and I is countable. For each i Î I, let X_i denote the linear span of {e₁, e₂, ¼, e_n}, then X = È_{i = 1}ⁿX_i. But the X_i are proper closed subspaces of X and so are nowhere dense, that is impossible by the Baire category theorem. Thus I is uncountable.
Let a Î X, a = å_{i Î I} l_i e_i where all l except finitely many are zero. Set ||a ||¢ = å_{i Î I} |l_i|. Then ||. ||¢ is obviously a norm on X. For i ¹ j , ||e_i - e_j ||¢ = 2 and I is uncountable, hence (X,||. ||¢) has no dense countable subset.