Differentiability of the operator norm My question is simple. Given finite-dimensional real Banach spaces $V, W$, is the operator norm on $\mathcal{L}(V, W) \setminus \{ 0 \}$ differentiable?
I know the standard Euclidean norm would be, but I don’t know what to do with this.

Consider $V=W=\mathbb R^n$ with $\|\cdot\|$ as the spectral norm on matrices (induced by the 2-norm on vectors). If $\|\cdot\|$ is differentiable, then for any $A,B$, the one-sided directional derivatives
$$\nabla_B\|A\|=\lim_{h\to0^+}\frac{\|A+hB\|-\|A\|}{h}$$
should satisfy $\nabla_B\|A\|=-\nabla_{-B}\|A\|$.
However, take $A=I$ and $B=\operatorname{diag}(1,0,\dots,0)$, and we have $\nabla_B\|A\|=1$ but $\nabla_{-B}\|A\|=0$.
