Linked Questions

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2answers
1k views

What is the sum of all real numbers from $0$ to $1$? [closed]

I wanted to know the approximate sum of real numbers from 0 to 1. Please tell me how we can find it.
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2answers
338 views

A question about an uncountable summation. [duplicate]

Possible Duplicate: The sum of an uncountable number of positive numbers Consider $\sum_{\lambda \in \Lambda} a_{\lambda}$ . Here all $a_\lambda $ is non-negative. Then I want to prove that if $\...
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2answers
121 views

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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1answer
476 views

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(...
9
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1answer
876 views

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 ...
7
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1answer
588 views

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 ...
6
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1answer
124 views

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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1answer
301 views

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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1answer
596 views

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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1answer
29 views

Meaning of absolute convergence when summing over countably infinite set.

I'm trying to understand the chapter on Elliptic functions in Stein's Complex Analysis. In particular, I am interested in the construction of Weierstrass's $\wp$ function. Let $\Lambda = \{n + m\tau :...
2
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1answer
142 views

Hilbert sum of $L_2(X_\nu,\mu_\nu)$ spaces.

Let $\{(X_\nu,\mu_\nu):\nu\in\Lambda\}$ be a family of measurable spaces. Is it true that $\bigoplus_2\{L_2(X_\nu,\mu_\nu):\nu\in\Lambda\}$ isometrically isomorphic to $L_2\left(\bigsqcup\{(X_\nu,\mu_\...
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1answer
135 views

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 ...
8
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0answers
264 views

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 $f$ ...
7
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0answers
294 views

Are uncountable “Schauder-like” bases studied/used?

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 $X$. ...
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0answers
45 views

Uncountably infinite series [duplicate]

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,...}...

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