Density of a semigroup in a Banach space

摘要

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) + ... + x(n), x(k) is an element of M, to be dense in X. We distinguish conditions under which the closure <(R(M))over bar> 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 is an element of X: f(x) >= 0}, f is an element of X*, and is minimal in the sense that every proper subarc of M lies in an open half-space {x is an element of X: f(x) > 0}, then <(R(M))over bar> - X. We apply our results to questions of approximation in various function spaces.