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Is it true, that in every concept of an infinite sum in a Banach space, encountered in an introductory functional analysis course, convergence is independent of the rearrangement of its terms ?

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    There's also a notion of summable families in a Banach space, which is a weaker property than absolute convergence, but which is still independent of rearrangement.2012-08-21

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$\mathbb R$ is a Banach space, and its concept of infinite sums is certainly not independent of rearrangements. The standard example is the alternating harmonic series $\sum_n \frac{(-1)^n}{n}$, which can be rearranged to have any limit you like, or none at all.