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Limit sets in normed linear spaces. (English) Zbl 1365.54025
Summary: 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 \(x_n\rightarrow 0\) 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.
MSC:
54F15 Continua and generalizations
46B20 Geometry and structure of normed linear spaces
40A05 Convergence and divergence of series and sequences
40A25 Approximation to limiting values (summation of series, etc.)
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