Polyak's theorem on Hilbert spaces

Abstract

We extend to infinite dimensional Hilbert spaces a celebrated result, due to B. Polyak, about the convexity of the joint image of quadratic functions. We show sufficient conditions which assure that the joint image is also closed. However, we prove that the closedness part of Polyak's theorem does not hold in general in the infinite dimensional setting. Finally, we give some applications to S-lemma type results.