Hi

Please, can some one give me lecture and exercises in Arzelą-Ascoli theorem ??

Thanks in advance!!

Miss_Lolitta

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- Apr 5th 2006, 03:04 PMmiss_lolittaArzelą-Ascoli theorem
Hi

Please, can some one give me lecture and exercises in Arzelą-Ascoli theorem ??

Thanks in advance!!

Miss_Lolitta - Apr 5th 2006, 06:40 PMThePerfectHackerQuote:

Originally Posted by**miss_lolitta**

- Apr 6th 2006, 02:56 AMRebesques
Why would a miss-"lolitta" be interested in such messy stuff? :eek:

Anyway. Here's one: Let be the set of Hölder continuous functions on , with exponent . Define the norm

.

Show that the closed unit ball of is compact in the space of all continuous functions on .

(Enjoy :rolleyes:.

Taken from Royden's "Real Analysis" ) - Apr 6th 2006, 06:17 AMmiss_lolitta
Really this is messy stuff!! :eek:

I wanted lecture in it because I have an assignment how to write a lecture with explaining by examples then represent this lecture in class..

But I have no ideas about this theorem?? :confused:

and How I perform this a simply.. without complexity?? :cool:

ThePerfectHacker & Rebesques....

Thanks so much :) - Apr 6th 2006, 10:22 AMRebesques
Well... If you want, you might say something like this. The Bolzano-Weirstrass theorem, sais that a bounded sequence converges. What can we say about a sequence of functions?

A problem is, we need to be precise about the mode of convergence - so consider the best possible, uniform convergence - the one implied by the sup-norm:

So, if we simply have a sequence of functions , and it is bounded (i.e., the sup-norm is finite for all), do we have uniform convergence? That is, is there a function with ???

No. The domain could be unbounded, and so our functions may not approach a limit function (though they are bounded).

We fix this (and alike problems) by demanding the domain be compact, say the interval

So, are we now to say uniform convergence is guaranteed? No, again. It could happen we only have pointwise convergence alone (not in the sup-norm, but only that for each , .)

The additional hypothesis to resolve this, is (uniform) equicontinuity; That is, the in the definition of continuity, holds globally for all and .

(Notice that that the Holder continuous functions in the last exercise do not satisfy this uniformly - but a relaxed form of the theorem can be used.)

So, every bounded and (uniformly) equicontinuous sequence of functions defined on , necessarily has a (uniformly) convergent subsequence - this is what the theorem says. :confused: :eek:

Example. Take . This is a bounded and equicontinuous sequence (show!). Note that for every does not exist (besides trivial cases).

But, by the Ascoli-Arzela theorem, there exists some subsequence and a function with

.

Hope that helps. - Apr 6th 2006, 12:45 PMmiss_lolitta
That's perfect!!

Well done!! ;)

Please,,Can you prove that :

Let K be a compact subset of Rp and let F be a collection of functions which continuous on K and have values in Rq. The following properties are equivalent:

1- The family F is bounded and uniformly equicontinuous on K.

2- Every sequence from F has a subsequence which is uniformly convergent on K.( this is another state for Arzelą-Ascoli theorem ) :confused:

Thanks :) soooo00S00oooo :) much - Apr 7th 2006, 11:18 AMRebesquesQuote:

Thanks soooo00S00oooo much

Quote:

Let K be a compact subset of Rp and let F be a collection of functions which continuous on K and have values in Rq. The following properties are equivalent:

1- The family F is bounded and uniformly equicontinuous on K.

2- Every sequence from F has a subsequence which is uniformly convergent on K.( this is another state for Arzelą-Ascoli theorem)

__1 implies 2__: This is the proof of the Arzela-Ascoli theorem. It is kind of tedious, but at least there's a diagonal argument.

a) Consider all points of with rational coefficients. Call this (numerable) set , and denote its elements by .

b) Let be any sequence of elements of . We show it has a pointwise convergent subsequence on .

The sequence is bounded; By the Bolzano-Weierstrass theorem, it has a convergent subsequence, say Apply the same argument for , yielding a convergent subsequence (note: the sequence is thus convergent at and ). Proceed in this fashion, to obtain a subsequence of the original sequence . Then, the sequence converges for all elements of .

c) We now show that converges uniformly on . That is, is Cauchy for all , in the uniform convergence norm.

Let (these will be fixed later) and . There exists some with

due to equicontinuity of the functions and the density of in . Also, we have

for sufficiently large .

Therefore

by (1) and (2). So

for all sufficiently large ; this means the subsequence is uniformly convergent on .

__2 implies 1__: Let be the metric space of continuous functions on endowed with the uniform convergence norm.

The given condition implies that is compact in , so it is**bounded**and closed in its topology.

We now show it is equicontinuous. Let belong to . Then there is a sequence in converging uniformly to .

Let (to be fixed later). Consider so large that

and, from the uniform continuity of , there exists

such that

for . We thus obtain

due to (3) and (4). So is uniformly equicontinuous. - Apr 7th 2006, 03:38 PMmiss_lolitta
Really I don't know how can I thank ..

I'm grateful for you

I repeat...

Thanks :) soooo00S00oooo :) much - Apr 7th 2006, 10:54 PMmiss_lolittaQuote:

Originally Posted by**Rebesques**

- Apr 10th 2006, 10:57 AMRebesquesQuote:

I'm so sorry

Not as sorry as I am.

Do come back with more theorems like this. - Apr 10th 2006, 12:58 PMmiss_lolitta
ok

click here if u sure :p

good luck :rolleyes: - Apr 10th 2006, 03:02 PMRebesques
That's nothing like the other one.

- Apr 11th 2006, 10:53 AMmiss_lolitta
I know that

but you understand it on the fly

try again to take bonus!! :D - Apr 12th 2006, 03:07 PMRebesquesQuote:

try again to take bonus!!

(...unless it's really much.) - Apr 13th 2006, 03:12 PMmiss_lolitta
oh,you can't be bribed..

but bonus was candy..Do you like this bribe?? ;)

Rebesques,,I would like to tell you that presentation I presented it to my professor who will say the result is it good or no on next Monday ,,

but I'm sure it's excellent!! :cool:

just because Rebesques helped me without any bribe..

regard..

lolitta