Math 524
Homework due 11/06/02
Reading: Section 3 Chapter 1 in Folland. Section 1 Chapter 2 in Folland.
Problem 1. (Prelim) Let fn: [0,1] →Rbe a sequence of continuously differentiable
functions (i.e. fn∈C1([0,1])) which satisfy fn(0) = 0, and
Z1
0
|f0
n(x)|2dx ≤1 for all n∈N.
Prove that there is a subsequence of fnwhich converges uniformly on [0,1].
Hint: recall that if a, b ≥0 and > 0 then 2ab ≤a2
+b2.
Problem 2. Let X={f: [0,1] →R;fpiecewise continuous}. We say that fis piecewise
continuous on [0,1] if fhas only finitely many discontinuities in [0,1]. For f , g ∈Xwe say
that f∼gis f−g≡0 on [0,1], except for finitely many points. This defines an equivalence
relation on X. Let Y={[f] : f∈X}where [f] denotes the equivalence class of f∈X. For
[f],[g]∈Ydefine
d([f],[g]) = Z1
0
|f−g|dx.
Show that ddefines a metric on Y. Show that the closed ball of radius 1 and center [0], i.e.
{[f]∈Y:d([f],[0]) ≤1}
is not compact.
Problems from Folland:
Chapter 1, Section 3: problems 9, 10.
Chapter 2, Section 1: problems 1, 2, 9.
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