During my calculations I've encountered a semidefinite program, e.g.
$$\max\{ \lVert X \rVert : \lVert X \circ M \rVert \le 1 \}$$
where $\circ$ denotes the entry-wise multiplication (this is called filtered $\gamma_2$ norm). In my case, the rows and columns of $X$ are indexed by the elements of a set $S$, which is a finite subset of the complex circle. Now I'm interested to extend the case to continuously infinite $S$ (always within the unit circle).
My question is: does the theory of semidefinite programming (or more generally, convex optimization) still hold over continuously infinite Hilbert spaces, or there are some results that hold only in the finite case? Any reference would be appreciated!
Since you wrote about duality, you might be interested in the concept of Fenchel duality. This holds even in Banach spaces (and there are also variants in topological vector spaces).