Re: Pointwise Convergence

On 3 Jan 2006 17:31:55 GMT, israel@xxxxxxxxxxx (Robert Israel) wrote:

>In article <96qkr19e2odt782as8shidpu28mufbqnr9@xxxxxxx>,
>David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx> wrote:
>>On Sun, 01 Jan 2006 15:44:07 EST, Ali Taghavi <alitghv@xxxxxxxxx>
>>wrote:
>>
>>>hi
>>>let K be a compact metric space,and C(K) be the space of all real valued
>>>continious maps defined on K. Is there a metric on C(K) for which the
>>>convergence of a sequence f_n is equivalent to pointwise convergence of f_n's
>>
>>In general, no. For example, no if K is uncountable.
>>
>>There _is_ a "_topology_ of pointwise convergence". If you look
>>at the description of this topology and assume that d is such
>>a metric you see that for every n there exists a finite set
>>F_n such that f = 0 on F_n implies d(f,0) < 1/n.
>
>Sorry, I don't see this.

Hmm, I guess I don't see it either, now that you point out
that nets are not sequences.

Ok. Let's consider the case K = [0,1] and show there is no
such metric in that case. The sort of sequence of functions
I had in mind does give a counterexample, the argument is
just a little different:

First, there exists a_1 in the interval (1/2, 1] such that
if f vanishes on [0,1/2] union [a_1, 1] then d(f,0) < 1.

(Proof: If not then there exists a sequence f_n such that
f_n vanishes on [1,1/2] union [1/2 + 1/n, 1] with
d(f_n, 0) >= 1. But that can't happen because for this
sequence we have f_n -> 0 at every point.)

Similarly there exists b_1 in (1/2, a_1) such that
if f vanishes on [0,b_1] union [a_1,1] then d(f,0) < 1/2.
Then there is a_2 in (b_1, a_1) such that if f vanishes
on [1,b_1] union [a_2,1] then d(f,0) < 1/3, and then
b_2 in (b_1, a_2) such that if f vanishes on [0,b_2]
union [a_2, 1] then d(f,0) < 1/4, etc.

In case some typos slipped in there: We're supposed
to have b_j increasing, a_k decreasing, and b_j < a_k
for all j,k.

Now choose p so that b_j < p < a_k for all j,k.
If f_n is a sequence such that f_n vanishes
on [0,p-1/n] union [p+1/n,1] then it follows
that f(f_n,0) -> 0. But such a sequence need
not tend to 0 at p, contradiction.

>A set U is a neighbourhood of 0 in the
>topology iff every _net_ converging pointwise to 0 is eventually in U,
>but the relation of d to pointwise convergence is only for sequences,
>not nets. Thus the answer to the analogous question "is there a
>metric on ell_1 for which the convergence of a sequence is equivalent
>to weak convergence of the sequence" would be yes: weak convergence of
>sequences in ell_1 is equivalent to norm convergence. But there is
>no finite set F_n of bounded linear functionals on ell_1 such that
>f(x) = 0 for f in F_n implies ||x|| < 1/n.
>
>Robert Israel israel@xxxxxxxxxxx
>Department of Mathematics http://www.math.ubc.ca/~israel
>University of British Columbia Vancouver, BC, Canada


************************

David C. Ullrich

.

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