Introduction to Analysis
Homework 12
1. (Rudin ex7.15) Suppose f is a real continuous function on R1, fn(t) = f (nt) for n = 1, 2, 3, · · · , and {fn} is equicontinuous on [0, 1]. What conclusion can you draw about f ? 2. (Rudin ex7.16) Suppose {fn} is an equicontinuous sequence of functions on a compact set K, and {fn} converges pointwise on K. Prove that {fn} converges uniformly on K.
3. (Rudin ex7.18) Let {fn} be a uniformly bounded sequence of functions which are Riemann- integrable on [a, b], and put
Fn(x) = Z x
a
fn(t)dt (a ≤ x ≤ b).
Prove that there exists a subsequence {Fnk} which converges uniformly on [a, b].
4. (Rudin ex7.19) Let K be a compact metric space, let S be a subset of C(K). Prove that S is compact if and only if S is uniformly closed, pointwise bounded, and equicontinuous.
(If S is not equicontinuous, then S contains a sequence which has no equicontinuous subsequence, hence has no subsequence that converges uniformly on K.)
5. Let X be a metric space and suppose that f : X → R1 has the following property: there are positive numbers M , α with 0 < α ≤ 1 such that
|f (p) − f (q)| ≤ M [d(p, a)]α
for any two points p, q in X. Then we say that f satisfies a H¨older condition on X with constant M and exponent α. Show that a family F of functions satisfying a H¨older condition with constant M and exponent α is equicontinuous.
6. (a) Let {fn} be a sequence of functions fn: (a, b) → R1 which are convex. Show that if {fn} converges pointwise on (a, b), then {fn} converges uniformly on each compact subset of (a, b).
(b) Let A be an open convex set in R2 and {fn} be a sequence of functions fn: A → R1 which are convex. Show that if {fn} converges pointwise on A, then {fn} converges uniformly on each compact subset of A.
7. Let {fn} be a sequence of functions fn : (a, b) → R1 which are convex. Suppose that {fn} are uniformly bounded. Show that {fn} contains a convergent subsequence and that the subsequence converges uniformly on each compact subset of (a, b).
