As far as I understand, there are Lipschitz functions $f:\mathbb{R}\to\ell^\infty$ that are nowhere differentiable in the Frechet sense. Where can I find such an example?



Non-differentiable Lipschitz functions
Define $f\_n(t)=\sin(nt)/n$ and $\phi(t)=(f\_1(t),\ldots)$. This has Lipschitz constant 1, but has no Fréchet derivative as far as I can tell. 


