Is a norm from any normed space $\mathbb{A}$ to $\mathbb{R}$ Frechet differentiable at zero? I've tried proving by contradiction that the norm is not Frechet differentiable at zero, but didn't get very far.
2025-01-12 23:58:22.1736726302
Differentiability of norm
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Consider the norm $|x|$ on $\mathbb R$. Clearly it is not differentiable at $0$.
An aside comment: Not exactly what you ask, but no norm on $\mathbb R^n$ (and so on any normed vector space) is differentiable everywhere, although you ask at zero. The same happens in fact to any distance on $\mathbb R^n$. See the relevant paper of Rosenholtz at http://www.jstor.org/stable/2320593.