Let $H$ be a Hilbert space and $\sum_k x_k$ a countable infinite sum in it. Lets say we partition the sequence $(x_k)_k$ in a sequence of blocks of finite length and change the order of summation only in those blocks, like this (for brevity illustrated only for the first two blocks $(x_1,\ldots,x_k,x_{k+1},\ldots,x_{k+l},\ldots )$ becomes
$(x_{\pi(1)},\ldots,x_{\pi(k)},x_{\gamma(k+1)},\ldots,x_{\gamma(k+l)},\ldots ),$ where $\pi$ and $\gamma$ are permutations.
If we denote the elements of the second sequence with $x'$, does anyone know, what will happen to the series $\sum _k x'_k$ in this case ? Can it stay the same ? Does staying the same requires additional assumptions ?