THE OPTIMAL LOWER BOUND FOR A POLYNOMIAL NORM WHICH IS A PRODUCT OF LINEAR AND CONTINUOUS FORMS IN A HILBERT SPACE

Abstract

The estimation of lower bounds for the norms of homogeneous polynomials which are products of linear forms in a Banach space, was obtained by K. Ball in a very precise description in the case where H is a complex Hilbert space with dimension ≥ n. He also managed to obtain a better bound estimate for c_n(H) = n^{−n/2}. The above result is taken as a corollary of Ball's theorem, which is not valid in the case of real Hilbert spaces. In this paper we studied the reasonable question if the above result is valid in the case of real Hilbert spaces.