What Topological Vector Spaces have an uncountable Schauder basis?

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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.
 
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  • #3
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The Schauder basis is countable by definition
Yeah, that’s why I put “uncountable Schauder basis” in quotes. Because the property that I(P) has is akin to a Schauder basis except it involves uncountable linear combinations rather than countable linear combinations.
 
  • #4
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I'm confused by the notation, if ##f:P\to F##, then what does ##f(a,b)## mean when ##a,b\in P##?

This looks pretty similar to a very uninteresting idea. If ##f## is a function ##A\to B## then ##f=\sum_{a\in A} f(a) \delta_a## where I haven't even told you that ##B## has an addition or a multiplication defined on it, but somehow this is still obviously true. I don't think this actually helps you do anything though.
 
  • #5
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I'm confused by the notation, if ##f:P\to F##, then what does ##f(a,b)## mean when ##a,b\in P##?

This looks pretty similar to a very uninteresting idea. If ##f## is a function ##A\to B## then ##f=\sum_{a\in A} f(a) \delta_a## where I haven't even told you that ##B## has an addition or a multiplication defined on it, but somehow this is still obviously true. I don't think this actually helps you do anything though.
Sorry, there’s a typo in my question, it should say that f is a function from Int(P) to F.

And yeah, after I posted my question I realized that if A is an uncountable set and F is a topological field, then the set X of functions from A to F is a topological vector space with the topology of pointwise convergence, and any f in X can be written uniquely as an uncountable unordered sum ##f=\sum_{a\in A} f(a) \delta_a##. My incidence algebra example was a special case of that.

So I guess I’m looking for examples of “uncountable Schauder bases” other than the topology of pointwise convergence. Any such example would have to be non-first countable, because in any first-countable topological vector space a convergent unordered sum can only have countably many nonzero terms.
 
  • #6
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So I guess I’m looking for examples of “uncountable Schauder bases” other than the topology of pointwise convergence. Any such example would have to be non-first countable, because in any first-countable topological vector space a convergent unordered sum can only have countably many nonzero terms.

It's not even totally clear to me what an unordered sum with uncountably many non zero terms means. With countably many it just means you can write it in any order and get a sequence that always converges to the same point, but how do you actually evaluate an uncountable sum?
 
  • #7
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It's not even totally clear to me what an unordered sum with uncountably many non zero terms means. With countably many it just means you can write it in any order and get a sequence that always converges to the same point, but how do you actually evaluate an uncountable sum?
It’s a standard notion in functional analysis. If ##X## is a topological vector space, ##\{x_i: i\in I\}\subseteq X##, and ##x\in X##, then we say that the unordered sum ##\Sigma_{i\in I}x_i## converges to ##x## if for every open set ##U## containing ##x##, there exists a finite set ##F\subseteq I## such that for every finite set ##J\subseteq I##, if ##F\subseteq J## then ##\Sigma_{i\in J}x_i\in U##. (There’s also an equivalent definition involving ##\Sigma_{i\in J}x_i -x## lying within neighborhoods of zero.)

Now there’s a theorem that if ##X## is a first-countable topological vector space (which includes normed vector spaces) and ##\Sigma_{i\in I}x_i## is convergent, then only countably many of the ##x_i##’s can be nonzero. But this need not be the case for non-first-countable spaces, like the topology of pointwise convergence I discussed earlier. So that is the context of my question about “uncountable Schauder bases”.
 

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