Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

Homework 8

Math 447: Real Variables

Exercise 1 [Ross, Ex. 21.8] Let (S, d) and (S*, d*) be metric spaces. Show that if f : S → S* is uniformly continuous, and if (sn) is a Cauchy sequence in S, then (f(sn)) is a Cauchy sequence in S*.

Exercise 2

(a) For each n ∈ N, let fn(x) = xn. Is the family of funcitons {fn(x)} uniformly equicontinuous on the interval [0, 1]? Justify your answer with a proof.

(b) For each r ∈ R+, let gr(x) = r2+3 sin(r)x. Is the family of funcitons {gr(x) | r ∈ R} uniformly equicontinuous on the interval (−∞, ∞)? Justify your answer with a proof.

Exercise 3 Prove Lemma 2 (and its converse) from Lecture 23:

Suppose X and Y are metric spaces, K ⊂ X compact, and F ⊂ C(K, Y ) is a family of continuous functions K → Y . Prove that F is equicontinuous if and only if F is uniformly equicontinuous.