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.
Best Answer
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.