Curves in Banach spaces which allow a $C^2$ parametrization

We give a complete characterization of those $f:[0,1]\to X$ (where $X$ is a Banach space which admits an equivalent Fr'echet smooth norm) which allow an equivalent $C^2$ parametrization. For $X=\mathbb{R}$, a characterization is well-known. However, even in the case $X={\mathbb{R}}^2$, several quite new ideas are needed. Moreover, the very close case of parametrizations with a bounded second derivative is solved.