Linked with [this question and discussion](
Bilinear product of two summable families), I am very
interested in counterexamples/results about the following questions (cf the end).
First, I recall that a
family $(a_i)_{i\in I}$ in a topological abelian group $(G,+)$ is called *summable* with sum $S$ iff
for all neighbourhood of zero $W$ it exists $J_W\subset_{finite} I$ such that for all $J$ with
$J_W\subset J\subset_{finite} I$, $(S-\sum_{i\in J}a_i)\in W$. It amounts to the same to say that the
net $J\mapsto \sum_{i\in J}a_i$ (from $2^{(I)}$, the set of finite subsets of $I$, ordered by
inclusion, to $G$) converges to $S$.

It is known that, when $I=\mathbb{N}$ (series $\sum_{n\geq 0}\,a_n$) the series
$\sum_{n\geq 0}\,a_n$ is summable iff it is unconditionaly convergent, i.e. the sequence of partial
sums
$$
N\to \sum_{n=0}^N\,a_{\sigma(n)}
$$

converges for all permutation $\sigma$ of $\mathbb{N}$.

**Question(s)** I am particularly interested in counterexamples/results about series
$\sum_{n\geq 0}\,a_n$ which are unconditionaly convergent but not absolutely convergent
in the following frameworks

- $K=[0,1]\subset \mathbb{R}$ and a series of continuous real functions $\sum_{n\geq 0}\,f_n$ unconditionaly convergent but not absolutely convergent i.e. $$ \sum_{n\geq 0}\,||f_n||_K<+\infty $$ (where $\|f\|_K=\sup_{s\in K}|f_s|$)
- $\mathcal{H}(\Omega)$ (space of holomorphic functions $\Omega\to \mathbb{C}$, where $\Omega\subset \mathbb{C}$ is not empty and open). In this context, absolutely convergent, for a series $\sum_{n\geq 0}\,f_n$, means that for all $K\subset_{compact} \Omega$, one has $$ \sum_{n\geq 0}\,||f_n||_K<+\infty $$

are there (counter-)examples or general results in these directions ?

Topological Vector Spacesand several other standard texts that touch on locally convex tensor products.) $\endgroup$ – Robert Furber Dec 17 '18 at 10:18