From: Joubert on 27 May 2010 13:36 Let Y be the set of continuous functions from [0,1] to R. Prove that the set: http://it.tinypic.com/r/2uqbw1t/6 is not precompact in Y, i.e. there is a sequence in K which has no converging subsequence in the max-norm
From: Stephen Montgomery-Smith on 27 May 2010 13:55 Joubert wrote: > Let Y be the set of continuous functions from [0,1] to R. Prove that the > set: > > http://it.tinypic.com/r/2uqbw1t/6 > > is not precompact in Y, i.e. there is a sequence in K which has no > converging subsequence in the max-norm Try some fairly simple piecewise linear functions, like f(x) = 2^n x if 0<x<2^(-n) = 1 otherwise.
From: Joubert on 27 May 2010 21:58 > Try some fairly simple piecewise linear functions, like > f(x) = 2^n x if 0<x<2^(-n) > = 1 otherwise. The derivative of this one is not continuous in 2^(-n) which makes he thing collapse. How can I smoothen it? And anyway this sequence seems to converge to the constantly 1 function with the max norm (but maybe I'm wrong here) so how does this make precompactness fail?
From: Robert Israel on 28 May 2010 02:48 Joubert <trappedinthecloset9985(a)yahoo.com> writes: > > > Try some fairly simple piecewise linear functions, like > > f(x) = 2^n x if 0<x<2^(-n) > > = 1 otherwise. > > The derivative of this one is not continuous in 2^(-n) which makes he > thing collapse. How can I smoothen it? Round off the corner a bit, e.g. using a parabolic piece. > And anyway this sequence seems to converge to the constantly 1 function > with the max norm (but maybe I'm wrong here) so how does this make > precompactness fail? Replace < by <=. It doesn't converge to 1 at x=0. -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
From: Joubert on 28 May 2010 14:28 Alright, thanks.
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