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I was reading section 1.46 of Rudin functional analysis.

He concluded E is equicontinous I do not know how? Please Help me

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Any Help will be appreciated

RobPratt
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Curious student
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1 Answers1

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This boils down to roughly the following: Given a family of smooth functions, a universal bound on their derivatives gives a universal Lipschitz constant which gives you equicontinuity.

Ciarán Ó Raghaillaigh
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  • Hi, May I ask how to prove "aa universal bound on their derivatives gives a universal Lipshcitz constant". For n=1, it can be proved by Mean Value Theorem. Is there an analogous MVT in multivariate functions? It will be extremely helpful if you can provide the related reference theorem in Baby Rudin. – Daaaaa Mar 31 '22 at 03:39
  • Oh, it is theorem 9.19 in Baby rudin? – Daaaaa Mar 31 '22 at 03:42
  • To apply theorem 9.19, we need to find an convex open set $E$. Is there a related theorem states that given $K_1 \subset K_2 \subset R^N$, which are both compact, there is an convex open set $E$ such that $K_1\subset E \subset K_2$? – Daaaaa Mar 31 '22 at 03:51
  • I find the answer for my question here, in case someone has the same confusion as I had. https://math.stackexchange.com/questions/1257553/a-multivariate-function-with-bounded-partial-derivatives-is-lipschitz – Daaaaa Mar 31 '22 at 04:01