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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?

Noah Schweber
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Keshav Srinivasan
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  • @NoahSchweber Yes, that’s exactly what it is. – Keshav Srinivasan Apr 07 '21 at 01:14
  • @NoahSchweber No, to be clear 1_[a,b] is the function which takes in an interval [c,d] and outputs 1 if a=c and b=d and outputs 0 otherwise. – Keshav Srinivasan Apr 07 '21 at 01:18
  • Yeah I just realized that. – Noah Schweber Apr 07 '21 at 01:18
  • Another example slightly generalising that of pointwise convergence is $B(H)$ equipped with the weak operator topology for $H$ a Hilbertspace. If $\{e_i\}_{i\in I}$ is an ONB then for any $A\in B(H)$ you have that: $$A=\sum_{ij} (Ae_j, e_i)\cdot e_i\otimes e_j^*$$ where convergence is in WOT - for a non-separable Hilbert space this sum is uncountable. This decomposition also works for the sub-algebras $\ell^\infty(\Bbb N)$ and $c_0(\Bbb N)$ of $B(H)$, where WOT becomes something like convergence that is uniform-on-countable-subsets. – s.harp Apr 15 '21 at 11:24

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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.

Jochen
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  • Yeah, I realized myself that basically any topology of pointwise convergence has this property. I’m looking for other examples though. – Keshav Srinivasan Apr 07 '21 at 15:10