Permutations of series in infinite-dimensional spaces
Let X be a linear space and \(A=\{a_ n\}_ 1^{\infty}\) a sequence of elements of X. Denote by S(A) the set of all \(x\in X\) having the form \(x=\sum^{\infty}_{n=1}a_{g(n)}\), where g is a permutation of the set \({\mathbb{N}}=\{1,2,...\}\). If X is finite-dimensional then S(A) is a linear variety [cf. \textit{E. Steinitz}, Bedingt konvergente Reihen und konvexe Systeme, J. Reine Angew. Math. 143, 128-175 (1913)]. In the case of infinite-dimensional X the author gives sufficient conditions for the set S(A) to be a closed linear variety.
scientific article; zbMATH DE number 4161338