Mixing
The weaker boundedness implies uniformly bounded
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Theorem (Arzelà-Ascoli). Eevery bounded equicontinuous sequence of functions in $C^{0}([a,b],mathbb{R})$ has a uniformly convergent subsequence.
The question asks to generalize the theorem with the weaker boundedness hypothesis. So, the theorem becomes
Theorem (Arzelà-Ascoli generalization). Every equicontinuous sequence $(f_{n})$ of functions with compact domain such that for $x in K$, $(f_{n}(x))$ is bounded in $C^{0}(K,mathbb{R})$ has a uniformly convergent subsequence.
So, I want to prove that
$$text{If $(f_{n}(x))$ is a bounded sequence for $x in K$, then $(f_{n})$ is uniformly bounded.}$$
I didnt get a good idea to start. Could someone give me just a hint?
equicontinuity arzela-ascoli
asked Jan 10 at 2:24
Lucas CorrêaLucas Corrêa
1,6151321
$endgroup$
add a comment |
$begingroup$
Theorem (Arzelà-Ascoli). Eevery bounded equicontinuous sequence of functions in $C^{0}([a,b],mathbb{R})$ has a uniformly convergent subsequence.
The question asks to generalize the theorem with the weaker boundedness hypothesis. So, the theorem becomes
Theorem (Arzelà-Ascoli generalization). Every equicontinuous sequence $(f_{n})$ of functions with compact domain such that for $x in K$, $(f_{n}(x))$ is bounded in $C^{0}(K,mathbb{R})$ has a uniformly convergent subsequence.
So, I want to prove that
$$text{If $(f_{n}(x))$ is a bounded sequence for $x in K$, then $(f_{n})$ is uniformly bounded.}$$
I didnt get a good idea to start. Could someone give me just a hint?
equicontinuity arzela-ascoli
asked Jan 10 at 2:24
Lucas CorrêaLucas Corrêa
1,6151321
$endgroup$
$begingroup$
The proof is outlined here: en.wikipedia.org/wiki/…
$endgroup$
– Ben W
Jan 10 at 2:31
add a comment |
$begingroup$
Theorem (Arzelà-Ascoli). Eevery bounded equicontinuous sequence of functions in $C^{0}([a,b],mathbb{R})$ has a uniformly convergent subsequence.
The question asks to generalize the theorem with the weaker boundedness hypothesis. So, the theorem becomes
Theorem (Arzelà-Ascoli generalization). Every equicontinuous sequence $(f_{n})$ of functions with compact domain such that for $x in K$, $(f_{n}(x))$ is bounded in $C^{0}(K,mathbb{R})$ has a uniformly convergent subsequence.
So, I want to prove that
$$text{If $(f_{n}(x))$ is a bounded sequence for $x in K$, then $(f_{n})$ is uniformly bounded.}$$
I didnt get a good idea to start. Could someone give me just a hint?
equicontinuity arzela-ascoli
asked Jan 10 at 2:24
Lucas CorrêaLucas Corrêa
1,6151321
$endgroup$
Theorem (Arzelà-Ascoli). Eevery bounded equicontinuous sequence of functions in $C^{0}([a,b],mathbb{R})$ has a uniformly convergent subsequence.
The question asks to generalize the theorem with the weaker boundedness hypothesis. So, the theorem becomes
Theorem (Arzelà-Ascoli generalization). Every equicontinuous sequence $(f_{n})$ of functions with compact domain such that for $x in K$, $(f_{n}(x))$ is bounded in $C^{0}(K,mathbb{R})$ has a uniformly convergent subsequence.
So, I want to prove that
$$text{If $(f_{n}(x))$ is a bounded sequence for $x in K$, then $(f_{n})$ is uniformly bounded.}$$
I didnt get a good idea to start. Could someone give me just a hint?
equicontinuity arzela-ascoli
equicontinuity arzela-ascoli
asked Jan 10 at 2:24
Lucas CorrêaLucas Corrêa
1,6151321
asked Jan 10 at 2:24
Lucas CorrêaLucas Corrêa
1,6151321
asked Jan 10 at 2:24
Lucas CorrêaLucas Corrêa
1,6151321
asked Jan 10 at 2:24
Lucas CorrêaLucas Corrêa
1,6151321
asked Jan 10 at 2:24
Lucas CorrêaLucas Corrêa
1,6151321
1,6151321
$begingroup$
The proof is outlined here: en.wikipedia.org/wiki/…
$endgroup$
– Ben W
Jan 10 at 2:31
add a comment |
$begingroup$
The proof is outlined here: en.wikipedia.org/wiki/…
$endgroup$
– Ben W
Jan 10 at 2:31
$begingroup$
The proof is outlined here: en.wikipedia.org/wiki/…
$endgroup$
– Ben W
Jan 10 at 2:31
$begingroup$
The proof is outlined here: en.wikipedia.org/wiki/…
$endgroup$
– Ben W
Jan 10 at 2:31
add a comment |
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$begingroup$
The proof is outlined here: en.wikipedia.org/wiki/…
$endgroup$
– Ben W
Jan 10 at 2:31