This article is cited in
20 papers
Density of a semigroup in a Banach space
P. A. Borodin M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract:
We study conditions on a set
$M$ in a Banach space
$X$ which are necessary or sufficient for the set
$R(M)$ of all sums
$x_1+\dots+x_n$,
$x_k\in M$, to be dense in
$X$. We distinguish conditions under which the closure
$\overline{R(M)}$ is an additive subgroup of
$X$, and conditions under which this additive subgroup is dense in
$X$. In particular, we prove that if
$M$ is a closed rectifiable curve in a uniformly convex and uniformly smooth Banach space
$X$, and does not lie in a closed half-space
$\{x\in X\colon f(x)\geqslant0\}$,
$f\in X^*$, and is minimal in the sense that every proper subarc of
$M$ lies in an open half-space
$\{x\in X\colon f(x)>0\}$, then
$\overline{R(M)}=X$. We apply our results to questions of approximation in various function spaces.
Keywords:
Banach space, additive semigroup, density, uniformly convex space, modulus of smoothness,
approximation, simple partial fractions.
UDC:
517.982.256+
517.538.5
MSC: 41A65,
46B20,
46B25 Received: 03.02.2014
Revised: 21.04.2014
DOI:
10.4213/im8220