Semnan UniversityInternational Journal of Nonlinear Analysis and Applications2008-68226220150813Polarization constant $mathcal{K}(n,X)=1$ for entire functions of exponential type354525210.22075/ijnaa.2015.252ENA. PappasCivil Engineering Department, School of Technological Applications, Piraeus University of Applied Sciences (Technological Education Institute of Piraeus), GR 11244, Egaleo, Athens, GreeceP. Papadopoulosadepartment of electronics engineering, school of technological applications, technological educational institution (tei)
of piraeus, gr 11244, egaleo, athens, greece.L. AthanasopoulouDepartment of Electronics Engineering, School of Technological Applications, Piraeus University of Applied Sciences (Technological Education Institute of Piraeus), GR 11244, Egaleo, Athens, GreeceJournal Article20141106In 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 generalizedĀ 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<br />$$<br />|L|=|widehat{L}|,;forall ;; L in<br />{mathcal{L}}^{s}(^{n}X);.<br />$$<br />If the previous relation holds for every $L in {mathcal{L}}^{s}left(^{n}Xright)$, then spaces ${mathcal{P}}left(^{n}Xright)$ andĀ $L in {mathcal{L}}^{s}(^{n}X)$ are isometric. We can also study the same problem using Fr$acute{e}$chet derivative.https://ijnaa.semnan.ac.ir/article_252_67988509b46f50477e7aba6e7d056fd0.pdf