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This is, surprisingly, rather like futurebird's post immediately below. I'd like to prove or disprove that if f_n is a sequence of functions in L^2[0,1] whose 2-norm goes to zero, then f_n(x) goes to zero for almost all x. I seem to remember that there's a counterexample, but I can't remember what, in fact, it is. Ideas, hints or blatant solutions? Thanks! |
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I'm trying to better understand convergence so I made upa problem for myself based on an example from class. I want to know if I'm answering my own questions correctly.
Define a sequence of functions fn(x) = 1 if x is in {r1, r2, ... , rn} and 0 otherwise. Where r1, r2, ... , rn are the first n rational numbers in some enumeration of all rational numbers. fn converges pointwise to the dirichlet function. But, can we say anything else about how fn --> f?
Uniform Convergence
Given e > 0 is there an N such that when n > N |fn - f | < e for all x? No. Just let e=.5 we cal alays find an x value where |fn - f | = 1. That is, a rational number that has not yet been listed by the time we reach n.
Convergence in Measure
Yes. The measure of the set where f and fn are not the same is *always* 0.
Almost Uniform Convergence
Yes. If we let A, the set of measure less the any e where uniform convergence fails be Q, the rationals I think we have almost uniform convergence. since m{Q}=0 < e for all e > 0.
Convergence in LP
Yes, the Lp norm of the fn and f is always 0 anyway.
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Dec. 26th, 2009 @ 10:21 am
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Долго искалось, и наконец-то нашлось слово
для oпределения русского неомoдернизма:
АНТИХУДОЖЕСТВЕННАЯ САМОДЕЯТЕЛЬНОСТЬ. (или -модерьмизм-...) |
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Dec. 25th, 2009 @ 05:41 pm
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Фернандо де Лука - сучасний италійський композитор-історицист та клавесиніст:
БОСТОНСЬКІ ЖЕРТВИ ТА ВТЕЧА ПОЛЯ РЕВЕРА.
Виконує автор. |
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Poll #1503463 How do you learn a new math subfield beyond your specialty?
Open to: All, detailed results viewable to: All, participants: 30How do you learn a new subject beyond your subfield? (Note that you can choose multiple answers)
The results of the poll are here. |
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So I'm reading Kanamori's book "The Higher Infinite" and a statement he deems to be too obvious to merit proof is giving me a bit of trouble. LJ perhaps isn't as busy as it used to be a few years ago, and considering that set theory is rather esoteric even within mathematics, I'm perhaps being a bit optimistic to post this here, but still, if you don't try you'll never know...
The statement in question is Lemma 11.8 (a).
( Read more... ) |
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Photos of the weekWe're back with more dazzling pictures from around the world. Congrats to ![[info]](http://l-stat.livejournal.com/img/userinfo.gif) marlenemcc, who has been awarded a virtual blue ribbon as the winner of our fourth photo contest. We hope you'll click over to LJ_Photophile poll and tell us your picks in pics!
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Curtains
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In LaTeX, is there way to define a custom continued fraction formula \mycfrac{an}{bn} with a single continuous zickzack fraction line?
I am not quite satisfied with this code
\textnormal{\LARGE K}_{n=1}^\infty \frac{\left.a_n\right|}{\left|b_n\right.}
which gives me  |
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Curtains
Thanks, again, for joining us. Until next time, stay snug!
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