Assume that $X$ is a Banach space. Is there a continuous map $f:X\to X$ such that $f$ is nowhere Frechet differentiable, but its restriction to every finite dimensional subspace is every where Frechet differentiable?



A (non trivial) continuous map on a Banach space which is nowhere Frechet differentiable
From <http://thaijmath.in.cmu.ac.th/index.php/thaijmath/article/viewFile/285/416>


Fr\'echet differentiability implies G\^ateaux differentiability, but the converse is true only for finite-dimensional Banach spaces, in general. As an example, the mapping $f:L^1[0,\pi]\rightarrow\mathbb{R}$ defined by $f(x)=\int\_0^\pi\mbox{sin}(x(t))dt$ is every where G^ateaux differentiable, but nowhere Fr\'echet differentiable [9]


[9] M. Sova, Conditions for differentiability in linear topological spaces, Czechoslovak Math. J. (Russian) 16 (1966) 339­-362.


