Re: Pointwise Convergence

On 5 Jan 2006 08:55:40 -0800, israel@xxxxxxxxxxx wrote:

>David C. Ullrich wrote:
>
>> 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:
>
>...
>
>Excellent! Much better than my proof. And here's a generalization
>to show there is no such metric if K is any uncountable compact
>metric space.

Fine. This "we leave the general case to the reader" really
works...

> I'll use p for the metric of K, to distinguish it from
>the
>metric d on C(K). Suppose d has the property that d(f_n, 0) -> 0 if
>and only if f_n -> 0 pointwise.
>
>I'll define sequences of subsets K_n of K and functions f_n in C(K)
>with the following properties:
>
>1) K_n is uncountable and compact
>2) K_n is a subset of K_{n-1}
>3) d(f_n, 0) < 1/n
>4) f_n >= 1 on K_n
>
>Take K_0 = K.
>Given K_{n-1}, for each point x of K_n

Was that supposed to be "for each point x of K"?

>there is some r(x) > 0 such that
>d(f,0) < 1/n for every f in C(K) that is 0 on
>x union {y in K: p(x,y) >= r(x)}.
>The set of balls B_{r(x)}(x) is an open cover of K_{n-1}, so there is a
>
>finite subcover F. At least one member of F, say B_{r(u)}(u), has
>uncountable intersection with K_{n-1}..
>Now we can take e > 0 such that
>K_n = {y in K_{n-1}: e <= p(u,y) <= r(u)-e}
>is uncountable. Define f_n(y) = g(p(u,y)) where
>g(t) = max(0, t (r(u)-t)/(e (r(u)-e))).
>It is easily seen that properties (1) to (4) are satisfied.
>
>Now since K_n are a nested sequence of nonempty compact sets,
>their intersection is nonempty. And since f_n >= 1 on this
>intersection, f_n does not go to 0 pointwise, although d(f_n, 0) -> 0
>as n -> infty. Thus d can't exist.
>
>On the other hand, for countable K, such a metric does exist.
>Namely, if K = {x_n: n in N}, let
>d_n(f, g) = max { |arctan(f(x_j)) - arctan(g(x_j))|: j <= n }
>and d(f,g) = sum_{n=1}^infty 2^{-n} d_n(f,g).
>
>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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