My Math Forum Showing that a space of functions is complete

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 March 29th, 2014, 11:59 AM #1 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.
 March 29th, 2014, 01:57 PM #2 Global Moderator   Joined: May 2007 Posts: 6,684 Thanks: 658 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

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