FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

1997, VOLUME 3, NUMBER 1, PAGES 37-45

Polynomial continuity

José G. Llavona

Abstract

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A mapping $f:\; X$® 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 $$e > 0 and bounded $B$Ì X, there are a finite set $\{p$₁, ¼ ,p_n} of polynomials on $X$ and $$d > 0 so that e whenever $x,y$Î B satisfy $|p$_j(x-y)| < d $(1$£ j £ n). Every compact (linear) operator is $P$-continuous. The spaces $L$¥[0,1], $L1[0,1]$ 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 $k$Î N $(k$³ 2) there is a $k$-homogeneous scalar valued polynomial on $\$\; \backslash ell\; \_1\; \$$ 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.

All articles are published in Russian.

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Last modified: November 16, 1999