If $P$ is an uncountable locally finite poset, then the incidence algebra $I(P)$ is a topological vector space (in fact a topological algebra) with the interesting property that every element $f$ can be written uniquely as an uncountable unordered sum $\Sigma_{a,b\in P: a\leq b}f(a,b)1_{[a,b]}$, i.e. $\{1_{[a,b]}:a,b\in P, a\leq b \}$ constitutes an "uncountable Schauder basis" for $I(P)$. I find this interesting because for normed vector spaces, a convergent unordered sum can only have countably many nonzero terms.
So my question is, what other topological vector spaces have convergent unordered sums with uncountably many nonzero terms? And also, what other topological vector spaces have this specific "uncountable Schauder basis" property, i.e. there exists an uncountable subset such that every element can be written as an unordered sum of scalar multiples of this subset, and such that there exists at least one convergent unordered sum of scalar multiples of this subset with uncountably many nonzero terms?
Maybe, this is just a special case (for the discrete order?) which looks more familiar: Take any set $I$ and the vector space $F(I)$ of real functions $I\to \mathbb R$. The topology of pointwise convergence is given by the seminorms $\|f\|_E=\max\{|f(x)|:x\in E\}$ with finite sets $E\subseteq I$. Then every $f\in F(I)$ has the representation $f=\sum\limits_{x\in I} f(x)\delta_x$, which is indeed a convergent series with respect to the semi-norms defined above.