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Let ##P## be an uncountable locally finite poset, let ##F## be a field, and let ##Int(P)=\{[a,b]:a,b\in P, a\leq b\}##. Then the incidence algebra $I(P)$ is the set of all functions ##f:P\rightarrow F##, and it's a topological vector space over ##F## (a topological algebra in fact) with an interesting property: every ##f\in I(P)## can be written uniquely as an uncountable unordered sum ##f=\Sigma_{a,b\in P:a\leq b}f(a,b)\delta_{a,b}## where ##\delta_{a,b}:P\rightarrow F## is defined by ##\delta_{a,b}(c,d)=1## if ##a=c## and ##b=d## and ##\delta_{a,b}(c,d)=0## otherwise. In other words, ##\{\delta_{a,b}:a,b\in P, a\leq b\}## is an "uncountable Schauder basis" for the topological vector space ##I(P)##. This is interesting because for normed vector spaces, convergent unordered sums can only have countably many nonzero terms, whereas the above unordered sum can have uncountably many nonzero terms.

So I'm wondering the following:

1. What other topological vector spaces have convergent unordered sums with uncountably many nonzero terms?

2. What other topological vector spaces have this kind of "uncountable Schauder basis", i.e. a uncountable subset B such that every element of the topological vector space can be written uniquely as an unordered sum of scalar multiples of elements of B, and such that there exists at least one convergent unordered sum of scalar multiples of elements of B with uncountably many nonzero terms? (That was a mouthful!)

Any help would be greatly appreciated.

Thank You in Advance.

So I'm wondering the following:

1. What other topological vector spaces have convergent unordered sums with uncountably many nonzero terms?

2. What other topological vector spaces have this kind of "uncountable Schauder basis", i.e. a uncountable subset B such that every element of the topological vector space can be written uniquely as an unordered sum of scalar multiples of elements of B, and such that there exists at least one convergent unordered sum of scalar multiples of elements of B with uncountably many nonzero terms? (That was a mouthful!)

Any help would be greatly appreciated.

Thank You in Advance.

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