Polyak's theorem on Hilbert spaces

TL;DR

This paper extends Polyak's theorem on the convexity of quadratic functions' joint image from finite to infinite dimensional Hilbert spaces, exploring conditions for convexity and closedness, with applications to S-lemma results.

Contribution

It generalizes Polyak's theorem to infinite-dimensional Hilbert spaces and analyzes conditions for convexity and closedness of quadratic functions' joint image.

Findings

Convexity of joint image extends to certain infinite-dimensional cases.

Closedness of the joint image generally fails in infinite dimensions.

Applications to S-lemma type results are provided.

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 give sufficient conditions which assure that the joint image is also closed. However, we show that, in general, the closedness part of Polyak's theorem does not hold in the infinite dimensional setting, even for quadratic functions generated by compact operators. We give some applications to S-lemma type results.