# Existence of solutions in optimization of Hilbert-space functionals over closed and bounded sets

Let $$\mathcal{H}$$ denote am infinite-dimensional Hilbert space of functions and let $$L(f)$$ denote a convex, continuous functional over $$\mathcal{H}$$. I would like to know under which conditions the optimization problem

$$\min L(f) \ s.t. ||f||_{\mathcal{H}} = 1,$$

has a solution in the closed and bounded set $$||f||_{\mathcal{H}} = 1$$. Obviously the unit ball is not compact in infinite-dimensional spaces. Could someone point me towards a related existence theorem? Thank you in advance.

Solutions do not exist in general under these assumptions: Let $$H=l^2$$, $$L$$ defined by $$L(f) = \sum_{k=1}^\infty k^{-1} f_k^2,$$ where $$f = (f_k)$$. This functional is convex and continuous, the infimum of $$L$$ on the unit sphere is $$0$$ (take $$f$$ equal to the unit vectors), but the infimum is not attained.
In order to get existence, some additional compactness needs to be present: $$L$$ having compact level sets, $$L$$ being completely continuous, etc.
• Can this be salvaged if one assumes that L is weakly lower semi-continuous? I would think that by the boundedness of $\{f, |f||=1\}$ and a subsequence argument, one can show that the infimum is attained. Commented Feb 12, 2020 at 12:26