In my (poor) attempt to answer Unconditional and absolute convergence in non-Banach spaces, essentially asking about whether or not we can get rid of the completeness assumption in the Dvoretzky-Rogers theorem (which states that a Banach space in which every unconditionally convergent series is absolutely convergent is finite-dimensional), I stumbled upon an answer on MathOverflow to a related question which at first seemed to me to be the perfect fit, though I failed to remember that absolute convergence only implies unconditional convergence in Banach spaces, for the rather obvious reason that "absolute convergence implies convergence" is a characterisation of complete spaces among normed spaces.
I then went to check the reference given by the answerer, Helmut H. Schaefer's Topological Vector Spaces, Section $10$: Nuclear spaces and absolute summability, in which lies the context of this question. I'll use the notations of Schaefer and quote the relevant results and remarks leading to my question for self-containment (and also due to the paywall/barrier of entry to access the text...). Sorry in advance for the long read.
$E$ will denote a locally convex space (l.c.s.) for the remainder of this post.
First, Schaefer's definitions of (unconditional) summability and absolute summability:
A family $\{x_\alpha: \alpha \in A\}$ in $E$ is called summable (Chapter III, Exercise $23$) if $\lim_H x_H$ exists in $E$, where $x_H = \sum_{\alpha \in H} x_\alpha$ and $H$ runs through the family of all finite subsets of $A$ directed by inclusion $\subset$; the limit $x \in E$ is then denoted by $\sum_{\alpha \in A} x_\alpha$, or briefly by $\sum_\alpha x_\alpha$. (If A is infinite, the limit can equivalently be taken along the filter of subsets of A with finite complement.)
A summable family $\{x_\alpha : \alpha \in A\}$ in $E$ is called absolutely summable if, for every continuous semi-norm $p$ on $E$, the family $\{p(x_\alpha) : \alpha \in A\}$ is summable in $\mathbb{R}$.
EDIT: as Eparoh noticed, the definition of absolute summability given here requires summability, unlike the definitions of absolute convergence and unconditional convergence in normed spaces.
Now, the definitions of the spaces $\ell^1(A, E)$ and $\ell^1[A, E]$ along with their topologies:
Now let $A$ be a fixed non-empty index set and let $E$ be a given l.c.s.; the set of all absolutely summable families $\mathbf{x} = \{x_\alpha: \alpha \in A\}$ in $E$ can evidently be identified with a subspace $S_a$ of the algebraic product $E^A$. Denote by $\mathfrak{U}$ any fixed base of convex circled $0$-neighborhoods in $E$, and by $r_U$ the gauge of $U \in \mathfrak{U}$; clearly, the mapping $$\mathbf{x} \longmapsto p_U(\mathbf{x}) = \sum_\alpha r_U(x_\alpha)$$ is a semi-norm on $S_a$, and the family of semi-norms $\{p_U: U \in \mathfrak{U}\}$ generates a topology under which $S_a$ is a l.c.s. that will be denoted by $\ell^1[A, E]$.
Similarly, the set of all summable families $\mathbf{x} = \{x_\alpha : \alpha \in A\}$ can be identified with a subspace $S$ of $E^A$, and for each $U \in \mathfrak{U}$ the mapping $$\mathbf{x} \longmapsto q_U(\mathbf{x}) = \sup\left\{\sum_\alpha |\langle x_\alpha, x'\rangle| : x' \in U^\circ\right\}$$ is a semi-norm on $S$; the family of semi-norms $\{q_U: U \in \mathfrak{U}\}$ generates a topology under which $S$ is a l.c.s. that will be denoted by $\ell^1(A, E)$. The obvious inclusion $S_a \subset S$ defines a canonical imbedding of $\ell^1[A, E]$ into $\ell^1(A, E)$ which is continuous, since $q_U(\mathbf{x}) \leq p_U(\mathbf{x})$ for each $\mathbf{x} \in S_a$ and $U \in \mathfrak{U}$.
My first concern is with the "obvious inclusion $S_a \subset S$", since, again, absolute convergence does not imply unconditional convergence in incomplete normed spaces, thus I'm worried that there is a fundamental difference in the definitions of summability and unconditional convergence (or absolute summability and absolute convergence) for $A = \mathbb{N}$ which makes Schaefer's theory (or more accurately Pietsch's, as Schaefer credits Pietsch for most of the results) not work in the context of the first linked post. Not relevant anymore!
Next, Theorem $10.7$ states:
Theorem. A locally convex space $E$ is nuclear if and only if the canonical imbedding of $\ell^1[\mathbb{N}, E]$ in $\ell^1(\mathbb{N}, E)$ is a topological isomorphism of the first space onto the second.
And its second and third corollaries read:
COROLLARY $2$. An $(F)$-space $E$ is nuclear if and only if every summable sequence in $E$ is absolutely summable. COROLLARY $3.$ A Banach space in which every summable sequence is absolutely summable is finite dimensional.
This is why I mentioned nuclear spaces in the title of this post, as this theorem appears to be a characterisation via absolute versus unconditional summability of nuclear spaces among all locally convex spaces. You'll note that Corollary $3$ is exactly the Dvoretzky-Rogers theorem, and Corollary $3$ is a consequence of Corollary $2$ and the fact that infinite-dimensional normed spaces cannot be nuclear.
Finally, the core of this question, the remark that immediately follows this theorem and its corollaries:
We remark in conclusion that the algebraic identity of the spaces $\ell^1[N, E]$ and $\ell^1(N, E)$ implies the identity of their respective topologies whenever $\ell^1[N, E]$ is infrabarreled (Exercise $36$); hence the completeness of $E$ is dispensable in Corollaries $2$ and $3$.
My question is then: what did Schaefer mean by "dispensable"? Does this mean that Corollary $3$ holds for all normed spaces, or does it mean something else, like "Corollary $3$'s statement can be adapted for normed spaces"? Or maybe, as said previously, I've misjudged the definitions of the summabilities, hence meaning that Corollary $3$ does hold for normed spaces but that for incomplete normed spaces it is not the same as the statement "A normed space in which every unconditionally convergent series is absolutely convergent is finite-dimensional"?
Feel free to edit or re-tag if necessary or appropriate.