[Math] infinite dimensional supporting hyperplane theorem

Question

I keep seeing an "infinite-dimensional separating hyperplane theorem" stated (e.g. page 8 of this document), but I can't find a corresponding version of the supporting hyperplane theorem.

Here's the separation theorem from the above link:
Two disjoint nonempty convex subsets of a topological vector space can be properly separated by a closed hyperplane (or continuous linear functional) if one of them has a nonempty interior.

So now consider the (open) epigraph of a convex real-valued function $f$ defined on an infinite-dimensional topological vector space. It seems like we should be able to conclude that there exists a supporting hyperplane to the epigraph at any point $(x, f(x))$. This meets the conditions of the above theorem: the epigraph is the convex set with the non-empty interior, while the point $(x, f(x))$ is the other set, which is convex, non-empty, and disjoint from the (open) epigraph.

This would tell us that a subgradient exists at any point.

Am I right about this? It seems right, but then again it's strange that I haven't been able to find it stated anywhere.

EDIT: I should have said "strict" epigraph. I took for granted that the epigraph formed by strict inequality was an open set, which was a mistake. Thankfully, this was clarified when I asked a related question.

Answer 1

This is the convex support theorem and usually listed as one of the many versions of Hahn-Banach. You can find it stated as theorems HB17 in Section 28.4 and HB4 in Section 12.31 of Schechter's Handbook of Analysis and its Foundations.

Some remarks:

The theorem HB4 states

Any convex function from a real vector space to $\mathbb{R}$ is the pointwise maximum of the affine functions that lie below it. That is, if $X$ is a real vector space and $p:X\to\mathbb{R}$ is convex, then for every $x_0\in X$ there exists $f:X\to\mathbb{R}$ affine that satisfies $f(x) \leq p(x)$ for all $x\in X$, and $f(x_0) = p(x_0)$.

and works for all (finite or infinite dimensional) vector spaces. The same theorem is also true if you work in the category of topological vector spaces. In particular, HB17 is simply a restatement of the above with "real vector space" replaced by "real topological vector space" and "function" replaced by "continuous function". The TVS version in fact follows immediately from the general version by noting that any convex function on a TVS that is bounded above on some open set is in fact continuous.