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### Positive Sums: Product

Lemma for: Trace Positive Sum Given the TAS $\overline{\mathbb{R}}_+$. For product sums: $$\omega:I\times J\to\overline{\mathbb{R}}_+:\quad{\sum}_{I\times J}\omega={\sum}_J{\sum}_I\omega$$ (...
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### A question about $\prod_{x\in \mathbb{R}^{*}}{x}$

When I was an elementary school student, I encountered some puzzles like "What's product of numbers of hair of each people in the world?", which answer was zero, because there's people with no hair(...
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### Transfinite series: Uncountable sums

If you sum an expression over an uncountable set $\sum_{x\in \mathbb{R}}f(x)$, then do we need $f(x)=0$ on all but a countable subset in order for the sum to have a finite value? If not can you give ...
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### Is Lebesgue integral w.r.t. counting measure the same thing as sum (on an arbitrary set)?

TL:DR; For arbitrary sets (not necessarily countable) we have two notions: Lebesgue integral w.r.t. the counting measure and sum of family indexed by this set. Are these two notions equivalent? For ...
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### Is there a sum of an uncountable set of Real numbers? [duplicate]

The addition of Real numbers is commutative, so instead of saying we can find the sum of a sequence $\{a_1,...,a_n\}$ of real numbers that are pairwise not equal, we can say that there is a sum of a ...
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### Convergence of Series of a Net's terms

I'm working through Dr. Pete Clark's convergence notes here: http://math.uga.edu/~pete/convergence.pdf and I've been thinking about Exercise 3.2.2 (a) and I am completely stumped. The exercise says ...
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### Is every Hilbert space an $L^2$ space?

Let $H$ be any Hilbert space. Must there exist a measure space $(X,\scr{M},\mu)$ such that we have a Hilbert space isomorphism: $$H\cong L^2(\mu)$$ Thank you
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### Is there a rigorous way to define uncountable products?

I'm dreaming of a way to define an uncountable product of real numbers. Of course any sensible definition should only converge for a sequence with only finitely many terms outside $[0, 1]$. It ...
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### Would an integral defined using partitions of an interval into infinitely many intervals make sense?

In the definition of Riemann integral or Darboux-integral we first study partitions (or tagged partition) of the given interval determined by finitely many points. To each partition and a function $f$ ...
We could define the following notion of basis in a way analogous to unconditional Schauder basis: If $X$ is a topological vector space over $\mathbb R$ and $B=\{b_i; i\in I\}$ be a subset of $X$. ...
I was fascinated when I heard that the most intuitive laws of arithmetic are no longer necessarily valid when it comes to the sum of an infinite sequence, which be denoted by S = \sum_{n=0,1,2,...}...