Let $f: \mathbb R_+ \times \mathbb R \to \mathbb R$. Under which assumptions does it make sense to compute $\frac{d}{dt}\Vert f(t,\cdot) \Vert_{L^\infty(\mathbb R)}$ and what is it?
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Article on supremum norm differentiability – mathcounterexamples.net Jul 16 '20 at 18:52
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The $L^p$ norm $1 < p < \infty$ is differentiable. – GEdgar Jul 16 '20 at 18:53
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@GEdgar What happens in the $\infty$ case? – Jun Jul 16 '20 at 19:00
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@Jun $\sup$ norm may not be differentiable on $\mathbb R^2$. A fortiori the $\sup$ norm is not differentiable on a space of real function. See here. – mathcounterexamples.net Jul 16 '20 at 19:13
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Theorem 3.1 of this Supremum norm differentiability article provides necessary and sufficient conditions for the $\sup$ norm to be differentiable at a point of $\mathcal C(T,E)$ the space of continuous functions from the topological space $T$ to the Banach space $E$.
This is interesting and leverage the notion of smooth point.
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