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| − | ==Practice Exam 4== | + | ==Practice Exam 4== |
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5. Let <math>(X,\mathcal{M}, \mu)</math> be a measure space with <math>0<\mu(X) < \infty</math>. Assume that <math>f_n \to f</math> <math>\mu</math>-a.e. and <math>\|f_n\|_p \leq M < \infty</math> for some <math>1<p<\infty</math>. If <math>1\leq r <p</math>, show that <math>f_n \to f</math>/math> in <math>L^r</math>. | 5. Let <math>(X,\mathcal{M}, \mu)</math> be a measure space with <math>0<\mu(X) < \infty</math>. Assume that <math>f_n \to f</math> <math>\mu</math>-a.e. and <math>\|f_n\|_p \leq M < \infty</math> for some <math>1<p<\infty</math>. If <math>1\leq r <p</math>, show that <math>f_n \to f</math>/math> in <math>L^r</math>. | ||
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4. For <math>n=1,2,\ldots</math>, let <math>f_n:I\to \mathbb{R}, I =[a,b]</math> be a subsequence of functions satisfying the following: If <math>\{x_n\}</math> is a Cauchy sequence in <math>I</math>, then <math>\{f_n(x_n)\}</math> is also a Cauchy sequence. Show that <math>\{f_n\}</math> converges uniformly on <math>I</math>. | 4. For <math>n=1,2,\ldots</math>, let <math>f_n:I\to \mathbb{R}, I =[a,b]</math> be a subsequence of functions satisfying the following: If <math>\{x_n\}</math> is a Cauchy sequence in <math>I</math>, then <math>\{f_n(x_n)\}</math> is also a Cauchy sequence. Show that <math>\{f_n\}</math> converges uniformly on <math>I</math>. | ||
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| + | ==later== | ||
| + | problem 3 on practice exams 7, 8, 9, and 10 | ||
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| + | problem 5 on practice exam 11 | ||
[[ 2010 Summer MA 598 Hackney|Back to 2010 Summer MA 598 Hackney]] | [[ 2010 Summer MA 598 Hackney|Back to 2010 Summer MA 598 Hackney]] | ||
Revision as of 04:54, 26 July 2010
Problems that we have not yet done
Practice Exam 4
5. Let $ (X,\mathcal{M}, \mu) $ be a measure space with $ 0<\mu(X) < \infty $. Assume that $ f_n \to f $ $ \mu $-a.e. and $ \|f_n\|_p \leq M < \infty $ for some $ 1<p<\infty $. If $ 1\leq r <p $, show that $ f_n \to f $/math> in $ L^r $.
Practice Exam 6
4. For $ n=1,2,\ldots $, let $ f_n:I\to \mathbb{R}, I =[a,b] $ be a subsequence of functions satisfying the following: If $ \{x_n\} $ is a Cauchy sequence in $ I $, then $ \{f_n(x_n)\} $ is also a Cauchy sequence. Show that $ \{f_n\} $ converges uniformly on $ I $.
later
problem 3 on practice exams 7, 8, 9, and 10
problem 5 on practice exam 11
