Unconditionally convergent series in some functional spaces

Question

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

1. $K=[0,1]\subset \mathbb{R}$ 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 $\|f\|_K=\sup_{s\in K}|f_s|$)
2. $\mathcal{H}(\Omega)$ (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 ?

Answer 1

A good resource for these things is Section IV.10 of Schaefer's Topological Vector Spaces, so you should look there for the proofs of the following statements. For $E$ a locally convex space, let $\ell^1[E]$ denote the set of absolutely summable ($\mathbb{N}$-indexed) series and $\ell^1(E)$ the set of unconditionally summable series. These spaces admit locally convex topologies such that the inclusion $\ell^1[E] \rightarrow \ell^1(E)$ is continuous, and in the case that $E$ is complete, $\ell^1[E] \cong \ell^1 \hat{\otimes} E$ and $\ell^1(E) \cong \ell^1 \check{\otimes} E$, where these are the projective and injective tensor products defined by Grothendieck.

It is then true that the inclusion mapping $\ell^1[E] \rightarrow \ell^1(E)$ is a linear homeomorphism iff $E$ is a nuclear space. This is actually Pietsch's sharpened version of Grothendieck's theorem. This answers (2), because $\mathcal{H}(\Omega)$ is a nuclear space for $\Omega$ open, so unconditional and absolute summability are the same.

If $E$ is infrabarrelled, for instance if $E$ is a normed space, then the mapping $\ell^1[E] \rightarrow \ell^1(E)$ is a linear homeomorphism iff it is surjective, so these spaces are nuclear iff unconditional and absolute summability coincide. Infinite-dimensional Banach spaces are not nuclear, so every infinite-dimensional Banach space contains an unconditionally convergent series that is not absolutely convergent. This fact was originally proven by Dvoretsky and Rogers, in quite a different way.

Therefore $C([0,1])$ contains an unconditionally summable series that is not absolutely summable. However, in this special case, it can be proved more easily. Just take your favourite non-absolutely but unconditionally summable series in a separable Banach space, such as $\left(\frac{e_n}{n}\right)_n$ in $\ell^2$. Any separable Banach space can be isometrically embedded as a closed subspace in $C([0,1])$, so the image of this series defines a series with the same properties in $C([0,1])$.

Answer 2

Actually, by a theorem of Dvoretzky and Rogers (P.N.A.S. 1950), every infinite dimensional Banach space admits series that are unconditionally convergent, but not absolutely convergent.

This can be found, e.g., in Lindenstrauss & Tzafriri: "Classical Banach Spaces I - Sequence Spaces" or Albiac & Kalton: "Topics in Banach space theory" or Li & Queffelec: "Introduction to Banach Spaces_Analysis and Probability" (Vol. 1.).