Rearrangement of series in Banach space and absolute convergence

Question

Let $X$ be a Banach space, and assume $\sum_{i=1}^\infty x_i$ and any of its rearrangements are convergent to the same value; do we have the conclusion that $\sum_{i=1}^\infty \|x_i\|$ is also convergent?

When $X=\mathbb{R}$ I think it is true, but for general Banach spaces, is it true?

Answer 1

No, this is not true. See for instance the following example, taken from [1] (Example 1.3.1).

Let $X=\ell_2$ be the Banach space of real sequences that are square summable, and consider the sequence $(x^{(k)})_{k\geq 1}\in X^{\mathbb{N}^\ast}$ defined by $$x^{(k)}_n = \begin{cases} 0 & \text{if } n\neq k \\ \frac{1}{n} &\text{otherwise.}\end{cases}$$ Then the series $\sum_{k=1}^\infty x^{(k)}$ converges to the sequence $$x = (1,\frac{1}{2},\frac{1}{3}, \dots, ,\frac{1}{n},\dots)\in X$$ for any rearrangement of its terms, but does not converge absolutely since $$\sum_{k=1}^\infty \lVert x^{(k)}\rVert_2 = \sum_{k=1}^\infty \frac{1}{k} = \infty.$$

Quoted from this same page:

A little bit down the road we shall see [...] that in every infinite-dimensional Banach space one can construct an unconditionally convergent series which is not absolutely convergent.

Now, the theorem "down the road" referred to is the Dvoretsky-Rogers Theorem, restated below (see e.g. [2]):

Theorem. (Dvoretsky-Rogers, 1950) If every unconditionally convergent series in a Banach space $X$ is absolutely convergent, then $X$ is finite dimensional.

(and it's not hard to see that in any finite Banach space every unconditionally convergent series is absolutely convergent; so this is an if, and only if.)

In short: for arbitrary Banach spaces, absolute convergent still implies unconditional convergence, but the converse is no longer true.

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[1] Kadets, Vladimir. Series in Banach Spaces: Conditional and Unconditional Convergence, 1997. DOI:10.1007/978-3-0348-9196-7

[2] Diestel, Joseph. The Dvoretsky-Rogers Theorem. Sequences and Series in Banach Spaces. Springer New York, 1984. 58-65. DOI:10.1007/978-1-4612-5200-9_6