I have read [(In French)](http://les.mathematiques.free.fr/pdf/chap10_11_12_13.pdf)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?



Differentials in different topologies
Let $V:=\mathcal S(\mathbb R)$, the space of smooth functions with rapid decay.  

Let $\gamma:\mathbb R\to V$ be the map which sends $t$ to the function $x\mapsto e^{-(x-t)^2}$.  

Then $\gamma$ is differentiable.


Pick a Hamel basis of $V$ which includes all the elements $\gamma(t)$, and which also includes all the elements $\gamma'(t)$.


Consider a linear automorphism $a:V\to V$ which is the identity on all the elements of the Hamel basis, except that it exchanges $\gamma'(0)$ and $\gamma'(1)$.


Then you can pull back the topology of $V$ by the map $a$. This gives a new topology on $V$.


When computed w.r.t that new topology, the derivative of $\gamma$ at $0$ is the element $\gamma'(1)\in V$.


