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In Theorem 10.27 of the book Functional Analysis, Calculus of Variations and Optimal Control, there is the following gradient formula:

gradient formula

($\operatorname{co}$ deotes the convex hull).

Is there an analogue of this result for functions $f:X\to\mathbb R$ on a general Banach (or at least Hilbert) space $X$ and the Fréchet derivative?

Here are the relevant definitions:

generalized gradient

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  • $\begingroup$ Have you tried adapting the proof of the said Theorem? $\endgroup$
    – leo monsaingeon
    May 10, 2020 at 21:38

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