FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
1997, VOLUME 3, NUMBER 1, PAGES 37-45
José G. Llavona
Abstract
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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 $k\in\mathbb{N}$
$(k\geq 2)$
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.
All articles are published in Russian.
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