Fréchet differentiable norms on spaces of countable dimension (Q1203522)

Using the Kadets-Klee renorming techniques it is proved as a main result that for each separable Banach space \(X\) and for each linear subspace \(L\) of countable algebraic dimension there is an equivalent locally rotund norm on \(X\) which is Fréchet-differentiable on \(L\setminus \{0\}\). In particular, any normed linear space of countable dimension admits a Fréchet-differentiable norm.