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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, is the space of all bounded functions . I want to show that is complete. It is easy to show that indeed it is, and furthermore that all sequences of functions converge uniformly to elements in . But is this strictly necessary? Should it not be sufficient to show that 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.


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