I am across two definitions for unconditional convergence for which it is not immediately obvious to me that they are equivalent. Here are the definitions. Throughout, $\frak{X}$ will denote a Banach space.
Definition 1.
Given a series $\sum x_n$ in $\frak{X}$, we say that this series converges unconditionally to $x$ if for every $\varepsilon>0$, there is a finite subset $J\subseteq \mathbb{N}$ such that for every finite subset $I$ such that $J\subseteq I\subseteq\mathbb{N}$ one has $\|x – \sum_{i\in I}x_i\|<\varepsilon$.
Definition 2.
A series $\sum x_n$ in $\frak{X}$ is said to converge unconditionally to $x$ if for any permutation $\sigma:\mathbb{N}\to \mathbb{N}$, the series $\sum x_{\sigma(n)}$ converges to $x$.
I was wondering how one might go about proving that these two definitions are equivalent.
Best Answer
This is proved in detail in:
In fact, Hildebrandt proves the equivalence of five properties A–E. Your definition 1 is his condition E and your definition 2 is his condition A.
Since Hildebrandt proves the equivalence of A and E (both directions) on page 960 and the paper is freely accessible it makes little sense to reproduce the clean and clear argument here.
Let me mention that condition A goes back to
W. Orlicz, Über unbedingte Konvergenz in Funktionenräumen (I), Studia Math. 4 (1933), 33–37.
while (according to Hildebrandt) condition E was studied by Moore in
E.H. Moore, General analysis, Memoirs of the American Philosophical Society, vol. 1, part 2, 1939, p. 63.