Given a Banach space $\mathcal{X}$, consider $\phi(u)=\|u\|_{\mathcal{X}}^2,$ is $\phi\in\mathcal{C}^2(\mathcal{X},\mathbb{R})$?
For a Hilbert space $\mathcal{H}$, we have $\left\langle\phi'(u),v\right\rangle=2(u,v)_{\mathcal{H}},\forall v\in \mathcal{H}$, but now what is the gateaux derivative of $\phi$ in a Banach space?
Usually, when we calculate the frechet derivative, we first guess the gateaux derivative, and then prove that the gateaux derivative is continuous, but now I even can't guess the gateaux derivative of $\phi$.
In general, it is not necessary to hold $\phi\in C^2(X,\mathbb R).$ For example, $X=C[a,b]$ with the $L^\infty$ norm.
However, if $X$ is reflexive, then there exists a differentiable norm which is equivalent to the original norm of $X$.