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JOURNALS // Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya // Archive

Izv. RAN. Ser. Mat., 2014 Volume 78, Issue 6, Pages 21–48 (Mi im8220)

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


 English version:
Izvestiya: Mathematics, 2014, 78:6, 1079–1104

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