Document Type : Research Paper

Authors

• A. Pappas ¹
• P. Papadopoulos ²
• L. Athanasopoulou ³

¹ Civil Engineering Department, School of Technological Applications, Piraeus University of Applied Sciences (Technological Education Institute of Piraeus), GR 11244, Egaleo, Athens, Greece

² adepartment of electronics engineering, school of technological applications, technological educational institution (tei) of piraeus, gr 11244, egaleo, athens, Greece.

³ Department of Electronics Engineering, School of Technological Applications, Piraeus University of Applied Sciences (Technological Education Institute of Piraeus), GR 11244, Egaleo, Athens, Greece

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 $L \in {\mathcal{L}}^{s}\left(^{n}X\right)$, 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