Some Ramsey type theorems for normed and quasinormed spaces
Volume 124 / 1997
Studia Mathematica 124 (1997), 81-100
DOI: 10.4064/sm-124-1-81-100
Abstract
We prove that every bounded, uniformly separated sequence in a normed space contains a "uniformly independent" subsequence (see definition); the constants involved do not depend on the sequence or the space. The finite version of this result is true for all quasinormed spaces. We give a counterexample to the infinite version in $L_p[0,1]$ for each 0 < p < 1. Some consequences for nonstandard topological vector spaces are derived.
