PROBLEM SET 5

1. Does {sin(nx)} converge in the L^{1} norm?

2. Give an example of a sequence of functions {f_{n}} which converges to the constant zero function in L^{1}, but so that f_{n}(x) does not converge to zero at any point of [0, 1].

3. If f_{n} → f in the L^{1} norm, show that there is a subsequence f_{n_{k}} which converges a.e. to f.

4. Prove that the set of continuous functions of compact support is dense in L^{1}.

5. For each 1 ≤ p ≤ ∞ give an example of a function which is in L^{p} but not in L^{q} for any q ≠ p.

6. If f ∈ L^{1} show that X_{f} = {g ∈ L^{1} : |g| ≤ |f|} is a compact set in the L^{1} topology (i.e., a sequence in this set has a subsequence converging to a point of the set).