Equivalent forms of unconditional convergence

Suppose (3) fails, and enumerate finite subsets of ${\mathbb{N}}_{\ge 1}$ by their finite codes. [given] Recursively build a listing as follows. At stage $k$, first append the least positive integer not yet listed, let $M$ be the greatest integer listed so far, and then take the least coded finite set $F_k\subseteq \{M+1,M+2,\dots \}$ with $\mathrm{\parallel }{\sum }_{n\in F_k}{x}_n\mathrm{\parallel }\ge \epsilon$ and append its members in increasing order. The negation of (3) supplies such an $F_k$ after every finite stage, and least codes make the recursion unique.

No integer is listed twice, because every block $F_k$ lies beyond all [given, L2, step 1.1] earlier entries. Every positive integer is eventually listed, since each stage appends the current least omitted integer. Thus the listing is a permutation of ${\mathbb{N}}_{\ge 1}$. Each $F_k$ is a consecutive block whose increment has norm at least $\epsilon$, so the rearranged partial sums are not Cauchy. By [L2], (1) therefore implies (3).

Assume (3). Given $\epsilon$, choose $N$ for $\epsilon \mathrm{/}2$. If finite [given, L1, step 2.1] $E,F$ both contain $\{1,\dots ,N-1\}$, their sums differ by two disjoint finite tail sums and hence by less than $\epsilon$. In particular, the ordinary partial sums are Cauchy, so [L1] gives a limit $s\in X$. Applying (3) once more to a finite $F$ containing a sufficiently long initial segment shows $\mathrm{\parallel }{\sum }_{n\in F}{x}_n-s\mathrm{\parallel }<\epsilon$. Thus the finite-subset net converges to $s$, and (3) implies (2).

Every permutation's initial index sets are cofinal among finite subsets: [given, L2, step 3.1] each fixed finite set is eventually included. Hence (2) makes every rearranged partial-sum sequence converge to the net limit. The ordinary initial segments are also cofinal, so this limit is the fixed-order sum. Thus (2) implies (1).

Under (3), any finite tail of any subseries is a finite tail set of the [given, L1, step 1.1, step 4.1] original series. It satisfies the Cauchy criterion, so [L1] proves (4). Conversely, if (3) failed, the union of the disjoint blocks $F_k$ from step 1.1, listed increasingly, would define a subseries having successive block increments of norm at least $\epsilon$, hence not Cauchy. Thus (3) and (4) are equivalent.

Assume (3), let $\mathrm{\mid }{\lambda }_n\mathrm{\mid }\le M$, and take a finite tail set $F$. A finite layer-cake decomposition shows that for $0\le t_n\le 1$, ${\sum }_{n\in F}{t}_n{x}_n$ is a convex combination of subset sums of $F$. Writing a real multiplier as its positive part minus its negative part, and a complex multiplier as the same decomposition of real and imaginary parts, gives

Steps 2.1--6.1 give both directions among all five conditions. The [given, step 4.1, step 6.1] common-sum identification is the conclusion of step 4.1.