Let $ f: X\to (-\infty,+\infty]$ that $ X$ is **an infinite dimensional space**.


What are the conditions for $f$ and space $X$ to have the following equality correct?


$$\partial f(x)=\{\nabla f(x)\}$$ for all $x\in X$


I know when space X is **finite dimensional** and $f$ is proper convex function then
$$\partial f(x)=\{\nabla f(x)\}$$ for all $x\in X$


Thanks.



the subdifferential at points of differentiability in infinite dimensional space
That depends on what you mean by $\nabla f$. If $X$ is a normed vector space and $f:X\to \mathbb{R}$ is Gâteaux differentiable, then $\partial f(x) = \{DF(x)\}$ for the Gâteaux derivative $DF(x)\in X^\*$ of $F$ at $x$; the proof is virtually identical to the finite-dimensional one using the definition of the Gâteaux derivative and the characterization of the convex subdifferential using the directional derivative (shameless plug: Theorem 4.5 of <https://arxiv.org/abs/1708.04180>).


If $\nabla F(x)\in X$ is the gradient, then this of course requires $X$ to be a Hilbert space.


