Showing that a space of functions is complete

king.oslo · March 29th, 2014, 10:59 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, 12:57 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, 10:59 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 11:09 AM.
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March 29th, 2014, 12:57 PM	#2
mathman Global Moderator Joined: May 2007 Posts: 6,586 Thanks: 612	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
$mathman is offline$

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