A single point as sum range of a series Can anyone give some clues to show that any infinite dimensional Banach space have a conditionally convergent series whose sum range is a single point?
Thanks in advance for any help.
 A: You can do this as follows. Find a total biorthogonal system in any separable infinite-dimensional subspace of the space (see Lindenstrauss-Tzafriri, Classical Banach Spaces, vol. 1, Section 1.f), say $\{x_i,x_i^*\}_{i=1}^\infty$ with $x_i\in X$, $x_i^*\in X^*$, $||x_i||=1$, $x_i^*(x_j)=\delta_{i,j}$. Introduce the series as a sum of the following differences (each bracket contains $i$ terms), ordered as one series.
$$\left(\frac1ix_i+\dots+\frac1ix_i\right)-\left(\frac1ix_i+\dots+\frac1ix_i\right).$$
Remarks: (1) Since $x_i^*$ of this sum is $0$, and $x_i^*$ of each of the other terms is $0$, each convergent rearrangement should converge to $0$ (since the system is total).
(2) If we open brackets without changing the order, we get a divergent rearrangement (the Cauchy criterion is not satisfied).
(3) If we rearrange each such sum as alternating, that is
$$\frac1ix_i-\frac1ix_i\dots+\frac1ix_i-\frac1ix_i,$$
we get a convergent rearrangement.
Note: There exist much more sophisticated methods of proving much stronger results, see Wojtaszczyk, Jakub Onufry, A series whose sum range is an arbitrary finite set.Studia Math. 171 (2005), no. 3, 261-281. 
