A series whose sum range is an arbitrary finite set

TL;DR

This paper demonstrates that in infinite-dimensional Banach spaces, it is possible to construct series whose sum range is any predetermined finite set, extending previous knowledge beyond the two-point case.

Contribution

It introduces a method to realize any finite set as the sum range of a series in infinite-dimensional Banach spaces, answering a question posed in 1984.

Findings

Any finite set can be realized as a sum range in Banach spaces.

The construction generalizes previous two-point examples.

It advances understanding of series behavior in infinite-dimensional spaces.

Abstract

In finitely-dimensional spaces the sum range of a series has to be an affine subspace. It is long known this is not the case in infinitely dimensional Banach spaces. In particular in 1984 M.I. Kadets and K. Wo\`{z}niakowski obtained an example of a series the sum range of which consisted of two points, and asked whether it is possible to obtain more than two, but finitely many points. This paper answers the question positively, by showing how to obtain an arbitrary finite set as the sum range of a series in any infinitely dimensional Banach space.