An extension of a corollary of the Arzela-Ascoli theorem for smooth functions
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I'm trying generalize this corollary for the case which the sequence of functions ${ f_n }_{n in mathbb{N}}$ are defined on a bounded domain (open and connected) $U subset mathbb{R}^m$ ($m geq 2$). I would like to know how I can prove this or if I need put some restrictions on more to this statement be true.
Thanks in advance!
proof-verification uniform-convergence arzela-ascoli
asked Dec 20 '18 at 19:40
GeorgeGeorge
872615
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add a comment |
$begingroup$
I'm trying generalize this corollary for the case which the sequence of functions ${ f_n }_{n in mathbb{N}}$ are defined on a bounded domain (open and connected) $U subset mathbb{R}^m$ ($m geq 2$). I would like to know how I can prove this or if I need put some restrictions on more to this statement be true.
Thanks in advance!
proof-verification uniform-convergence arzela-ascoli
asked Dec 20 '18 at 19:40
GeorgeGeorge
872615
$endgroup$
add a comment |
$begingroup$
I'm trying generalize this corollary for the case which the sequence of functions ${ f_n }_{n in mathbb{N}}$ are defined on a bounded domain (open and connected) $U subset mathbb{R}^m$ ($m geq 2$). I would like to know how I can prove this or if I need put some restrictions on more to this statement be true.
Thanks in advance!
proof-verification uniform-convergence arzela-ascoli
asked Dec 20 '18 at 19:40
GeorgeGeorge
872615
$endgroup$
I'm trying generalize this corollary for the case which the sequence of functions ${ f_n }_{n in mathbb{N}}$ are defined on a bounded domain (open and connected) $U subset mathbb{R}^m$ ($m geq 2$). I would like to know how I can prove this or if I need put some restrictions on more to this statement be true.
Thanks in advance!
proof-verification uniform-convergence arzela-ascoli
proof-verification uniform-convergence arzela-ascoli
asked Dec 20 '18 at 19:40
GeorgeGeorge
872615
asked Dec 20 '18 at 19:40
GeorgeGeorge
872615
edited Dec 31 '18 at 23:15
George
asked Dec 20 '18 at 19:40
GeorgeGeorge
872615
asked Dec 20 '18 at 19:40
GeorgeGeorge
872615
asked Dec 20 '18 at 19:40
GeorgeGeorge
872615
872615
add a comment |
add a comment |
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After do some research, I found this lecture notes, which pratically solves the problem as we can see by the proofs of the lemma and of the proposition $2$ of this notes. Applying the Arzela Ascoli theorem a countable number of times as it was applied on proposition $2$, we develop a countable collection of subsequences of our original sequence and remains apply a standard diagonal argument (the same argument which is used on the proof of Arzela Ascoli theorem) in order to ensure that the uniform limit of the sequence $(f_n)$ is differentiable, i.e., if $f_n rightarrow f$, then $f in mathcal{C}^{infty}(U)$.
answered Dec 31 '18 at 23:16
GeorgeGeorge
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1 Answer
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1 Answer
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$begingroup$
After do some research, I found this lecture notes, which pratically solves the problem as we can see by the proofs of the lemma and of the proposition $2$ of this notes. Applying the Arzela Ascoli theorem a countable number of times as it was applied on proposition $2$, we develop a countable collection of subsequences of our original sequence and remains apply a standard diagonal argument (the same argument which is used on the proof of Arzela Ascoli theorem) in order to ensure that the uniform limit of the sequence $(f_n)$ is differentiable, i.e., if $f_n rightarrow f$, then $f in mathcal{C}^{infty}(U)$.
answered Dec 31 '18 at 23:16
GeorgeGeorge
872615
$endgroup$
add a comment |
$begingroup$
After do some research, I found this lecture notes, which pratically solves the problem as we can see by the proofs of the lemma and of the proposition $2$ of this notes. Applying the Arzela Ascoli theorem a countable number of times as it was applied on proposition $2$, we develop a countable collection of subsequences of our original sequence and remains apply a standard diagonal argument (the same argument which is used on the proof of Arzela Ascoli theorem) in order to ensure that the uniform limit of the sequence $(f_n)$ is differentiable, i.e., if $f_n rightarrow f$, then $f in mathcal{C}^{infty}(U)$.
answered Dec 31 '18 at 23:16
GeorgeGeorge
872615
$endgroup$
add a comment |
$begingroup$
After do some research, I found this lecture notes, which pratically solves the problem as we can see by the proofs of the lemma and of the proposition $2$ of this notes. Applying the Arzela Ascoli theorem a countable number of times as it was applied on proposition $2$, we develop a countable collection of subsequences of our original sequence and remains apply a standard diagonal argument (the same argument which is used on the proof of Arzela Ascoli theorem) in order to ensure that the uniform limit of the sequence $(f_n)$ is differentiable, i.e., if $f_n rightarrow f$, then $f in mathcal{C}^{infty}(U)$.
answered Dec 31 '18 at 23:16
GeorgeGeorge
872615
$endgroup$
After do some research, I found this lecture notes, which pratically solves the problem as we can see by the proofs of the lemma and of the proposition $2$ of this notes. Applying the Arzela Ascoli theorem a countable number of times as it was applied on proposition $2$, we develop a countable collection of subsequences of our original sequence and remains apply a standard diagonal argument (the same argument which is used on the proof of Arzela Ascoli theorem) in order to ensure that the uniform limit of the sequence $(f_n)$ is differentiable, i.e., if $f_n rightarrow f$, then $f in mathcal{C}^{infty}(U)$.
answered Dec 31 '18 at 23:16
GeorgeGeorge
872615
answered Dec 31 '18 at 23:16
GeorgeGeorge
872615
answered Dec 31 '18 at 23:16
GeorgeGeorge
872615
answered Dec 31 '18 at 23:16
GeorgeGeorge
872615
872615
add a comment |
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