# Metric Spaces

1. Mar 5, 2009

### tomboi03

Let X be a set, and let fn : X $$\rightarrow$$ R be a sequence of functions. Let p be the uniform metric on the space Rx. Show that the sequence (fn) converges uniformly to the function f : X $$\rightarrow$$ R if and only if the sequence (fn) converges to f as elements of the metric space
(RX, p)

I'm not sure how to do this problem...

Can someone help me out?

Thanks!

2. Mar 5, 2009

### yyat

Where are you stuck? Start by writing down all the relevant definitions (uniform metric, uniform convergence, convergence in metric spaces) and you will be almost done.

3. Mar 7, 2009

### de_brook

please define what is mean't by " uniform metric"

4. Mar 7, 2009

### yyat

It's the metric obtained from the http://en.wikipedia.org/wiki/Uniform_norm" [Broken], so

$$d(f,g)=\sup_{x\in X}|f(x)-g(x)|$$

where $$f,g:X\to\mathbb{R}$$.

Last edited by a moderator: May 4, 2017