Re: Difficult question; Kuratowski embedding, wodyjslawski theorem

In article
<6372235.1146671915176.JavaMail.jakarta@xxxxxxxxxxxxxx
orum.org>,
melanie <melanie@xxxxxxxxx> wrote:

(Wojdyslawski Thorem)
3.The image h(X) of the Kratowski embedding
h: X -> S is closed subset of
convex hull Z in S defined by h(X).

Write a member of Z as z = sum_{j=1}^m t_j h(b_j)
where b_j in X, t_j > 0, sum_j t_j = 1.
Hint: If this is the limit of a sequence h(a_n),
show d(b_j, a_n) -> 0 as n -> infinity.


What do you mean by "if this is the limit of a
sequence h(a_n)"?
Do you mean z is the limit of h(a_n) i.e.
h(a_n) -> sum_{j=1}^m t_j h(b_j) ?

Yes.

Could you explain more?

You want to show h(X) is closed in Z, so you consider
a member z of Z that is the limit of a sequence in
h(X),
and show that z is in h(X).

I tried to prove h(a_n) -> sum_{j=1}^m t_j h(b_j).
Something like,

|sum_{j=1}^m t_j h(b_j) - h(a_n)|
=<|sqrt(sum (t_j)^2)*sqrt(sum (h(b_j))^2) -h(a_n)|
=|sqrt(sum (h(b_j))^2) - h(a_n)|
(Using Cauchy-Shwartz)

but this does not seem like to go through.

What inequality should I use to show?




Robert Israel
israel@xxxxxxxxxxx
Department of Mathematics
http://www.math.ubc.ca/~israel
University of British Columbia Vancouver,
BC, Canada
.

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