Week 6

Uniform convergence

Exercise 6.1 [Homework 3, Problem 2(c)]

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. Show that $B(X,Y)$ is complete if $Y$ is.

(Here $B(X,Y)$ denotes the set of bounded functions from $X$ to $Y$ endowed with the metric $d_{B(X,Y)}(f,g)=\underset{x\in X}{sup}{d}_Y(f(x),g(x))$.)

Exercise 6.2 [Homework 3, Problem 5]

Let $\{f_n{\}}_{n=1}^{\mathrm{\infty }}$ be a sequence of functions from a metric space $(X,d_X)$ to a metric space $(Y,d_Y)$ and $f:X\to Y$ be a function. Prove that $f_n\to f$ uniformly if and only if $d_Y(f_n(x_n),f(x_n))\to 0$ for every sequence $\{x_n{\}}_{n=1}^{\mathrm{\infty }}\subseteq X$.