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March 29th, 2014, 11:59 AM   #1
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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,

Last edited by king.oslo; March 29th, 2014 at 12:09 PM.
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March 29th, 2014, 01:57 PM   #2
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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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