# Differentials under different topologies

I have read (In French)that the differential of a function depends on the topology and not the norm, the latter is rather easy to grasp, the first is hard for me to construct.

Norms being equivalent in finite dimensional spaces, I am looking for an example (in a functional space maybe) to showcase to what extent differentials may be just so different depending on the initial topology (implied by two different non-equivalent norms). Could someone suggest a classic example thereof?

• Even continuity of a function on a topological vector space will depend on the topology. Consider for instance the linear functional $L: f\mapsto f'(0)$ on the vector space $C^1((-1,1))$, where the latter is equipped with either the $C^0$-norm or the $C^1$-norm. In the first case, $L$ is discontinuous, hence, non-differentiable, and in the second, $L$ is its own differential. Dec 8 '17 at 1:10
• @MoisheCohen indeed, easy and simple example thank you for that. Dec 8 '17 at 8:31