Argue that for every normed space $\mathbb{X} \neq \{ 0 \}$ the norm map $\| \ldotp \|_\mathbb{X} : \mathbb{X} \to \mathbb{R}$ is not Fréchet differentiable at $0$.
Not really sure where to start on this question. I know the absolute value function is not Fréchet differentiable at $0$.
Best Answer
If it were Frechet, then there would be a (bounded) linear operator $A$ so that $$ \|v\|=A(v)+o(\|v\|). $$ Now insert $-v$ and compare.