Isometries between spaces of real polynomials of degree at most \(n\) (Q2814319)

The paper under review considers surjective isometries between the spaces $\mathcal P_A(^{\leq n} E)$ and $\mathcal P_A(^{\leq n} F)$ of approximable polynomials of degree at most $n$ defined on real Banach spaces $E$ and $F$, and endowed with the usual sup-norm on the unit ball. It is proved that every isometric isomorphism $T:\mathcal P_A(^{\leq n} E)\to \mathcal P_A(^{\leq n} F)$ has the form $T(P)=\pm \overline P \circ s^t\circ J_F$, where $J_F$ is the canonical embedding of $F$ into $F''$, $s$ is an isometric isomorphism of $E'$ onto $F'$, and $\overline P$ is the Aron-Berner extension of $P$ to $E''$. The proof is based on transposing $T$, identifying the dual of the spaces of approximable polynomials with spaces of integral polynomials $\mathcal P_I(^{\leq n} F)$, and using the description of the extreme points of these spaces. \par It is worth mentioning that the extreme points of the space $\mathcal P_I(^{\leq n} F)$ have the form $\pm \sum_{j=0}^n y^j$, when $ y$ varies over all points in the closed unit ball of $F''$ [\textit{C. Boyd} and \textit{A. Brown}, Rev. Mat. Iberoam. 33, No. 4, 1149--1171 (2017; Zbl 1426.46028)]. This causes a difference with the case of the space of $n$-homogeneous integral polynomials, whose extreme points have the form $\pm x^n$ when $x$ varies over all points in the unit sphere of $F''$ [\textit{C. Boyd} and \textit{R. A. Ryan}, J. Funct. Anal. 179, No. 1, 18--42 (2001; Zbl 0977.46017)]. This difference justifies the interesting extra work made by the authors in the case of non-homogeneous polynomials. \par The paper also deals with isometric isomorphisms between spaces of integral polynomials of degree at most $n$ and spaces of all continuous polynomials of degree at most $n$. In these cases, conditions involving the Radon-Nikodým property or the approximation property are considered.