Let $X$ be a topological vector space (you can assume $X$ is locally convex if it makes any difference). Let $C \subset X$ be nonempty, and let $f$ be a continuous linear functional on $X$.
When does $f$ attain a minimum on $C$?
Obviously, if $C$ is compact, then it contains a minimizer of $f$ (and the linearity of $f$ is not required for this fact). But is there anything else we can say?
I am also familiar with the fact that lower semicontinuous functions can be minimized over compact sets, but here I am asking about properties of $C$ that guarantee continuous linear functionals can be minimized.
Added Theorem 2.32 in this book is interesting. It says that if $X = \mathbb R^n$ and $f$ is continuous and coercive, i.e. $$\lim_{\| x \| \to \infty}f(x) = \infty,$$ then $f$ attains a minimum on $C$ provided only that $C$ is closed. I'm not sure if the proof will generalise to an infinite dimensional normed TVS.
Check out Corollaries 3.22 and 3.23 in the book “Functional Analysis, Sobolev Spaces and Partial Differential Equations” by H. Brezis. They talk about convex lower-semicontinuous functions.
Edit: The first one, for instance, implies that $f\in X^*$ attains its minimum if $C$ is convex, bounded and closed. But to be honest, I wouldn’t know how to generalize it to locally convex topological vector spaces.