Re: Representation of Analytic Functions
- From: "G. A. Edgar" <edgar@xxxxxxxxxxxxxxxxxxxxxxxxxxx>
- Date: Fri, 03 Mar 2006 08:23:32 -0500
In article
<22538649.1141325833854.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>, Maury
Barbato <mauriziobarbato@xxxxxxxx> wrote:
map from R^n (the
No. You are essentially trying to find a surjective
space of all c_n's) to the space of analyticfunctions (on the real
line, I presume). However, the latter space isinfinite dimensional
(e.g. the functions 1,x,x^2,... are all linearlyindpendent). So it will
be impossible to find a surjective map onto it fromthe finite
dimensional space R^n.
Why?
What you say is true, but "infinte dimensional" is
not the reason.
Are you saying that the answer to my original question
is No?
There is no continuous map. R^n is sigma-compact, but the space of
analytic functions is not.
There is a surjective map from the one-dimensional
space [0,1]
onto the infinite-dimensional space [0,1]^T, with T
countably infinite.
--
G. A. Edgar
http://www.math.ohio-state.edu/~edgar/
My Best Regards,
Maury
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
.
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