Metric Spaces

Let X be a set, and let fn : X [tex]\rightarrow[/tex] 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 [tex]\rightarrow[/tex] 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!
 
316
0
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.
 
please define what is mean't by " uniform metric"
 
316
0
please define what is mean't by " uniform metric"
It's the metric obtained from the http://en.wikipedia.org/wiki/Uniform_norm" [Broken], so

[tex]d(f,g)=\sup_{x\in X}|f(x)-g(x)|[/tex]

where [tex]f,g:X\to\mathbb{R}[/tex].
 
Last edited by a moderator:

The Physics Forums Way

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top