From: Joubert on
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
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

> 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
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
Alright, thanks.