``` wu :: forums (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi) riddles >> putnam exam (pure math) >> Complete set (Message started by: wonderful on Sep 8th, 2008, 4:19pm) ``` Title: Complete set Post by wonderful on Sep 8th, 2008, 4:19pm Let S be the set of all continous function on [0,1], with norm ||f(x)||= integral |f(x)| from 0 to 1. Show that S is not complete.Have A Great Day! Title: Re: Complete set Post by Obob on Sep 8th, 2008, 11:12pm The norm should be ||f||= int |f|.  The completion of C[0,1] with respect to this norm is the space L^1[0,1] of all measurable functions f such that int |f| < infty, with norm given by ||f||=int |f|.  For instance, the function g given by g(x) = 0 for x<1/2 and g(x) = 1 for x>=1/2 is the limit of a sequence of continuous functions with respect to this norm (just take a sequence of piecewise linear functions which start off at 0, shoot up to 1 more and more steeply near 1/2, and stay at 1 from then on).  Since C[0,1] with this norm is a sub-normed space of L^1[0,1] and since there is a sequence in C[0,1]  converging to an element of L^1[0,1]-C[0,1], C[0,1] cannot be complete. Title: Re: Complete set Post by wonderful on Sep 16th, 2008, 4:09pm Excellent Obob!Have A Great Day! Powered by YaBB 1 Gold - SP 1.4! Forum software copyright © 2000-2004 Yet another Bulletin Board