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I have been thinking about some conjectures of mine regarding non-uniformly convergent sequences of functions. I'm happy to discuss my work so far, if anyone wants to know, but for brevity let me just say that I am particularly interested in finding a reference which discusses some theory of non-uniformly convergent sequences of functions, specifically for functions from R2->R. Does anyone know of a good place to look for this?.

I have been trying to find a reference today for this subject and could not, is that because nobody thinks it's very interesting? I tried google, wiki, googlebooks, ect... but there are SO many ref's for uniform convergence that come up, it's hard to find what I'm looking for.

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closed as too broad by Michael Albanese, Claude Leibovici, BlackAdder, hardmath, Mark Fantini Dec 30 '14 at 7:01

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs. If this question can be reworded to fit the rules in the help center, please edit the question.

I don't follow. Do you want sequences that converge, just not uniformly? In what topology? There are plenty of other possible topologies to consider, and that's what functional analysis is for. –  Qiaochu Yuan Aug 24 '10 at 22:51
I want sequences which converge, and the convergence should be non-uniform. I'm just using the standard euclidean norm. The neat thing about these sequences is that they can converge to different functions in different regions of the phase plane. e.g. f(n) = n^2x/n^2(x+1) goes in the limit to 1/x unless x=0, then its just the zero function. stuff.mit.edu/afs/athena/course/18/18.06-esg/18.034/notes/… –  Matt Calhoun Aug 24 '10 at 23:09
So, pointwise convergence? If you use that keyword you should find plenty of references. (You should specify this. There are lots and lots of ways for sequences of functions to converge, and pointwise convergence is one of the most useless.) –  Qiaochu Yuan Aug 24 '10 at 23:31
As for your conjectures, you can post them as questions if you want. The answers may be well-known. –  Qiaochu Yuan Aug 24 '10 at 23:34
@Matt, if a sequence converges pointwise, it converges to one function—although the limit function may not be writable with one equation. –  Mariano Suárez-Alvarez Aug 24 '10 at 23:53

1 Answer 1

The reason why your search was unsuccessful was explained in a comment by paul garrett:

Pointwise convergence preserves few properties of functions. Measurability. Not continuity, not differentiability, ... and monotone and dominated convergence theorems show that pointwise convergence by itself is insufficient to assure that the limit of the integrals is the integral of the limit. To say that a function is a pointwise limit of very nice functions says almost nothing about the limit function. Thus, the "uselessness" of the concept. (The mystery of its dominant role in standard first-year "analysis" classes is a separate matter!) :)

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