Polarization constant \mathcal{K}(n,X)=۱ for entire functions of exponential type

In this paper we will prove that if L is a continuous symmetric n-linear form on a Hilbert space and \widehat{L} is the associated continuous n-homogeneous polynomial, then ||L||=||\widehat{L}||. For the proof we are using a classical generalized  inequality due to S. Bernstein for entire functions of exponential type. Furthermore we study the case that if X is a Banach space then we have that|L|=|\widehat{L}|,  \forall   L \in{\mathcal{L}}^{s}(^{n}X).If the previous relation holds for every L \in {\mathcal{L}}^{s}\left(^{n}X\right), then spaces {\mathcal{P}}\left(^{n}X\right) and  L \in {\mathcal{L}}^{s}(^{n}X) are isometric. We can also study the same problem using Fr\acute{e}chet derivative.

Keywords:

Polarization constants , polynomials on Banach spaces , polarization formulas