Polynomial continuity

J. Llavona

Fundamentalʹnaâ i prikladnaâ matematika, Tome 3 (1997) no. 1, pp. 37-45.

Résumé

A mapping $f\colon\,X\to Y$ between Banach spaces $X$ and $Y$ is said to be polynomially continuous ($P$-continuous, for short) if its restriction to any bounded set is uniformly continuous for the weak polynomial topology, i.e., for every $\varepsilon>0$ and bounded $B\subset X$, there are a finite set $\{p_1,\ldots,p_n\}$ of polynomials on $X$ and $\delta>0$ so that $\|f(x)-f(y)\|\varepsilon$ whenever $x,y\in B$ satisfy $|p_j(x-y)|\delta$ $(1\leq j\leq n)$. Every compact (linear) operator is $P$-continuous. The spaces $L^\infty [0,1]$, $L^1[0,1]$ and $C[0,1]$, for example, admit polynomials which are not $P$-continuous. We prove that every $P$-continuous operator is weakly compact and that for every $k\in\mathbb N$ $(k\geq2)$ there is a $k$-homogeneous scalar valued polynomial on $\ell_1$ which is not $P$-continuous. We also characterize the spaces for which uniform continuity and $P$-continuity coincide, as those spaces admitting a separating polynomial. Other properties of $P$-continuous polynomials are investigated.