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Jan
7
comment Generalization of Tonelli's Theorem for Series
There's a definition for the sum of uncountably many non-negative reals here: math.stackexchange.com/questions/106102/… -- as the sup over all the finite sums. Also well known that for this sum to be finite, there must be no more than a countable number of non-zero elements.
Jan
2
revised Generalization of Tonelli's Theorem for Series
corrected small hole in my proof; added a tag
Dec
28
comment Generalization of Tonelli's Theorem for Series
A good question. I'm studying from a book and no definition is given. Yes either set could be uncountable, and I was a bit sloppy there, because the sum could be finite if enough of them are zero (but that just dumps you into the countable case). If it is the 3rd, I don't think there is even anything to be proved. I was assuming the left side was the sum of individual elements indexed by A x B, which might be your first definition. I don't know what your second definition means. I didn't tag it as measure theory because we haven't defined meansures yet, but that is where we are headed..
Dec
28
asked Generalization of Tonelli's Theorem for Series
Dec
25
comment Uncountable sets on $(0,\infty)$ also uncountable on $[b,\infty]$ for some $b > 0$
Very helpful, Arthur.
Dec
25
comment Uncountable sets on $(0,\infty)$ also uncountable on $[b,\infty]$ for some $b > 0$
Gregory Grant and basket are saying the same thing of course, and I didn't think of it that way That surely nails it.
Dec
25
comment how to show strictly increasing function on an interval has continuous inverse
The word "surjective" has not previously come up in this discussion. However, the comment from whzecomjm illuminates the matter. We have to be careful about the range in which "surjective" applies.
Dec
25
asked Uncountable sets on $(0,\infty)$ also uncountable on $[b,\infty]$ for some $b > 0$
Nov
23
awarded  Nice Answer
Aug
5
awarded  Yearling
Dec
31
awarded  Nice Answer
Nov
22
comment Does L'Hôpital's work the other way?
Expand the top and bottom in a Taylor's polynomial. If the first term of both polynomials is zero, then you look at the second term. If both of those are zero, you go on to the third term, etc. Sooner or later you get a term which is not zero, and that is the one which shows you the limit.
Aug
5
awarded  Yearling
Feb
20
answered Functions and Infinitesimals?
Feb
20
comment “Ito-Riemann” integration
have you checked out this question: math.stackexchange.com/questions/426786/…
Jan
23
comment Existence of a Bound Related to Erratically Converging Sequences?
Certainly once you pick $\epsilon$ you must be able to find an N so that for n > N the elements are within $\epsilon$ of the limit. For every sequence you have to pick N big enough to achieve that. I believe my construction meets that requirement, because the jumps are getting smaller with each $2^n$. So for any given $\epsilon$ the jumps of $2^{-n}$ will eventually (for large enough n) put the entire tail of the sequence in the epsilon neighborhood of the limit; but you will still get all those erratic elements -- more and more as $n \rightarrow \infty$.
Jan
23
comment $n^{th}$ root of a matrix.
@mtiano Why would the existence of the root depend on the sign of the eigenvalues? Are we not admitting complex numbers? Also, if the matrix is not invertible the nth roots are still there even though those entries are zero; and the matrix is not expressible in exponential form. As per answer below, I was moving towards Jordan form -- if you can find the roots of a diagonalizable matrix, so too can you find the roots of the matrix in Jordan form.
Jan
22
comment $n^{th}$ root of a matrix.
Continue with your diagonalization idea. If the matrix is diagonalizable can you say something about its roots?
Jan
19
comment Integral Operator Theory on $L^2[0,1]$
Would like to help you be warmer, etc. but I actually couldn't quite make my way thru your notation. Can you rewrite this with latex? See this site for assistance:physicsforums.com/showpost.php?p=3977517&postcount=3
Jan
17
answered How do we prove the error estimation of the rectangle method