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I have come across the fact: In a Banach space, if a series $\sum_{n=1}^{\infty}x_n$ is absolutely summable in $X$, then it is unconditionally summable in $X$. So, I was searching for the definition of unconditionally convergent and got it in this post: Equivalent definitions of unconditional convergence

One of the two above definitions is also mentioned in Wikipedia. Here it is: https://en.wikipedia.org/wiki/Unconditional_convergence

But in Wiki, I got another equivalent definition that I should mention below:

A series $\sum_{n=1}^{\infty}x_n$ is unconditionally convergent if for every sequence $\{\epsilon_n\}_{n=1}^{\infty}$ with $\epsilon_n\in{-1,1}$, the series $$\sum_{n=1}^{\infty}\epsilon_nx_n$$ converges.

Here my questions are:
(a) Can anyone please explain why the above definition is equivalent to the former definitions?
(b) Can you please suggest a clean proof of this statement "In a Banach space, if a series $\sum_{n=1}^{\infty}x_n$ is absolutely summable in $X$, then it is unconditionally summable in $X$." I have gone through many proofs in stack exchange related to this fact but didn't get one satisfying. (Suggestion of a link will also be very useful).
Thank you beforehand!!!

user1551
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Pro_blem_finder
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    For (a), it seems that there is a proof in the book by Christopher Heil referenced in the wikiwand page that you linked, or at least in the expanded edition which is referenced in the french version of the wikiwand page https://books.google.fr/books?id=prfuUT0Sw-AC&pg=PA97#v=onepage&q&f=false – Gribouillis Nov 23 '24 at 08:47
  • @Gribouillis, Thank you, very much. It seems I got the answers to both questions. We already know any absolutely summable sequence in Banach space is summable. So, in the book, " Christopher Heil, A Basis Theory Primer "(in English) Page 94, Theorem 3.10, the condition (e) of the theorem will be satisfied and the theorem shows the equivalency with the statement of unconditionally convergent. I think this is enough to complete the proof of (b). – Pro_blem_finder Nov 23 '24 at 09:18

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