Showing that a space of functions is complete

king.oslo · March 29th, 2014, 11:59 AM

Hello there,

$(X, \rho)$ is the space of all bounded functions $f : K \to \mathbb{R}$ .

I want to show that $X$ is complete. It is easy to show that indeed it is, and furthermore that all sequences of functions converge uniformly to elements in $X$ . But is this strictly necessary? Should it not be sufficient to show that $f_n \to f$ pointwise? Is the uniform convergence critical to the completeness of X? And if yes, why?

Thank you for your time.

Kind regards,
Marius

mathman · March 29th, 2014, 01:57 PM

In general, completeness in any metric space means convergence using the metric. The space of bounded functions uses a sup norm, so that means you need to show uniform convergence.

March 29th, 2014, 11:59 AM	#1
king.oslo Senior Member Joined: Sep 2010 From: Oslo, Norway Posts: 162 Thanks: 2	Showing that a space of functions is complete Hello there, $(X, \rho)$ is the space of all bounded functions $f : K \to \mathbb{R}$ . I want to show that $X$ is complete. It is easy to show that indeed it is, and furthermore that all sequences of functions converge uniformly to elements in $X$ . But is this strictly necessary? Should it not be sufficient to show that $f_n \to f$ pointwise? Is the uniform convergence critical to the completeness of X? And if yes, why? Thank you for your time. Kind regards, Marius Last edited by king.oslo; March 29th, 2014 at 12:09 PM.
$king.oslo is offline$

March 29th, 2014, 01:57 PM	#2
mathman Global Moderator Joined: May 2007 Posts: 6,626 Thanks: 622	In general, completeness in any metric space means convergence using the metric. The space of bounded functions uses a sup norm, so that means you need to show uniform convergence. Thanks from MarkFL and king.oslo
$mathman is offline$

Thread Tools
$Show Printable Version$ Show Printable Version $Email this Page$ Email this Page
Display Modes
$Linear Mode$ Linear Mode $Hybrid Mode$ Switch to Hybrid Mode $Threaded Mode$ Switch to Threaded Mode

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
Question on a vector space of functions	student42	Real Analysis	2	August 10th, 2013 06:59 AM
Space trapped between functions	OriaG	Calculus	3	August 29th, 2012 12:56 PM
space of p-summable sequences complete?	e161065	Real Analysis	3	April 25th, 2010 04:49 AM
Space of bounded functions	saxash	Abstract Algebra	0	November 24th, 2009 04:02 AM
Vector functions & space curves	remeday86	Calculus	1	February 16th, 2009 05:43 PM