Two distinct differentials at the same point, with respect to different norms

Question

Is it possible for a function $f:E\to F$ (where $E,F$ are normed vector spaces) to admit two distinct differentials in a point $a\in E$ ? Of course one for every pair of norms on $E$ and $F$. It is known that if the two norms on $E$, and the other two of $F$ are equivalent, then it is not possible. So an example must be found on an infinite dimensional space (like a space of functions). Can you suggest such an example?

Answer 1

Sure. Let $F$ have infinite dimension and let $x_1,x_2,\dots\in F$ be linearly independent. We can then pick two different norms for which this sequence of linearly independent vectors converges to two different vectors. Say $\|\cdot\|_1$ is a norm on $F$ such that the sequence $(x_n)$ converges to $x$ and let $\|\cdot\|_2$ be a norm on $F$ such that $(x_n)$ converges to $y$, where $x\neq y$. Define $f:\mathbb{R}\to F$ by $f(1/n)=x_n/n$ if $n$ is a positive integer, interpolating linearly in between these values, defining $f(t)$ arbitrarily for $t>1$, $f(0)=0$, and $f(t)=-f(-t)$ for $t$ negative. Then it is easy to see that $f$ is differentiable at $0$ with respect to both norms on $F$, but the derivative $f'(0)$ with respect to $\|\cdot \|_1$ is $x$ while with respect to $\|\cdot\|_2$ it is $y$.