Limit sets in normed linear spaces
Volume 147 / 2017
Colloquium Mathematicum 147 (2017), 35-42
MSC: Primary 54F15, 46B20; Secondary 40A05, 40A25.
DOI: 10.4064/cm6868-5-2016
Published online: 21 November 2016
Abstract
The sets of all limit points of series with terms tending to 0 in normed linear spaces are characterized. An immediate conclusion is that a normed linear space $(X,\| \cdot \| )$ is infinite-dimensional if and only if there exists a series $\sum x_n$ of terms of $X$ with $x_n\to 0$ whose set of limit points contains exactly two different points of $X$. The last assertion could be extended to an arbitrary (greater than 1) finite number of points.
