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
Practice exam 12, numbers 2, 3, and 4c
