Polarization constant $mathcal{K}(n,X)=1$ for entire functions of exponential type

Abstract

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 $L in {mathcal{L}}^{s}left(^{n}Xright)$, then spaces ${mathcal{P}}left(^{n}Xright)$ and $L in {mathcal{L}}^{s}(^{n}X)$ are isometric. We can also study the same problem using Fr$acute{e}$chet derivative.