Linked Questions

23
votes
3answers
4k views

Does uncountable summation, with a finite sum, ever occur in mathematics?

Obviously, “most” of the terms must cancel out with opposite algebraic sign. You can contrive examples such as the sum of the members of R being 0, but does an uncountable sum, with a finite sum, ...
12
votes
2answers
2k views

Can sum of a series be uncountable

There are several methods to say whether sum of series is finite or not. Can we say whether sum of series is countable or not. For example $S_n=\Sigma_{0 \leq i \leq n}{2^i}= 2^{n+1}-1$ So for $n=\...
18
votes
4answers
3k views

Can we add an uncountable number of positive elements, and can this sum be finite?

Can we add an uncountable number of positive elements, and can this sum be finite? I always have trouble understanding mathematical operations when dealing with an uncountable number of elements. ...
7
votes
2answers
812 views

Why do we distinguish between infinite cardinalities but not between infinite values?

More specifically, why are we "allowed" to denote $|\mathbb{N}|<|\mathbb{R}|$ but not $\sum\limits_{n\in\mathbb{N}}1<\sum\limits_{r\in\mathbb{R}}1$? Can we distinguish between "countable ...
15
votes
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
votes
4answers
732 views

Is there a continuous version of the Borel-Cantelli lemma?

Given a sequence of events $A_n$ for $n\in \mathbb N$, the first Borel Cantelli lemma states that, if the sum over all probabilities $\sum_{i=1}^n P(A_n)$ is finite, then the probability of the limit ...
11
votes
2answers
2k views

Does 'uncountable sequence' make sense?

In the question Can the natural number have an uncountable set of subsets? the poster uses the phrase 'uncountable sequence'. Does that phrase make sense? No one in the question seems to object, but ...
31
votes
1answer
900 views

We have sums, series and integrals. What's next?

We know how to sum or average a finite number of terms: sums. We know how to sum a countable infinite number ${\beth_0}$ of terms: series. We know how to sum ${\beth_1}$ terms: integrals. How to ...
8
votes
2answers
287 views

Sum of a series indexed by ordinals

If $\mu$ is an ordinal, how can we formalize that $$ \sum_{\lambda<\mu}x_{\lambda}=z $$ When $\mu=\omega$, this is just the usual infinite series, the partial sums converge to $z$. What is the ...
7
votes
1answer
581 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 ...
1
vote
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 $\...
3
votes
0answers
575 views

Generalization of Tonelli's Theorem for Series

Let $A, B$ be sets and $x_{n,m}$ $n \in A, m \in B$ be a doubly infinite sequence of extended non-negative reals indexed by A and B. Show that $\sum_{(n,m) \in A \times B} (x_{n,m})$ = $\sum_{n \in ...
4
votes
2answers
162 views

Prob. 10, Sec. 3.5, in Kreyszig's Functional Analysis: How to show that this set is at most countable?

This is Prob. 10, Sec. 3.5, in the book Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $X$ be an inner product space, let $M$ be an uncountable orthonormal subset of $X$...
2
votes
1answer
243 views

Is it possible to make a sum of uncountable series of elements of a group or a ring?

Groups, rings and fields are equipped with binary operations. These can be applied repeatedly: $a_1+a_2+a_3+\dots$ to produce a sum of many elements, perhaps countably many. Can this be done also ...
0
votes
1answer
279 views

Are Hilbert-Schmidt operators in non-separable Hilbert spaces compact?

The definition of Hilbert-Schmidt operator should still be valid even when the Hilbert space is not separable: If $e_i$ for $i\in I$ is an orthonormal basis for a Hilbert space, and $\mbox{Trace}(T)=...

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