From: Stephen Montgomery-Smith
Subject: Re: 2 monomorphisms = 1 isomorphism?
Date: Sat, 04 Dec 1999 21:49:51 -0600
Newsgroups: sci.math
Keywords: A Schroeder-Bernstein theorem for Banach spaces? (no)
"Volker W. Elling" wrote:
>
> Hi everybody,
>
> this problem has been bothering me for a long time now:
>
> Given two things A,B so that
> A is isomorphic to a subthing B' of B and
> B is isomorphic to a subthing A' of A,
> is A isomorphic to B?
>
> Here, things can be sets, vector spaces, topological spaces, groups,
> whatever you like. Isomorphisms are bijective mappings that preserve
> thing structure in some sense.
>
> I know the statement is true for
> ++ sets (no structure, so any bijective mapping is an isomorphism;
> Schroeder-Bernstein theorem),
> ++ vector spaces (take a basis of A', extend it to a basis of A,
> f applied to the basis of A yields a basis for B', extend that one to
> a basis of B, g applied to the basis of B yields a basis for A':
> dim(A) = dim(B') <= dim(B) = dim(A') <= dim (A), so dim(A)=dim(B). )
>
> However, I would like to have something general, in the fashion of
> category theory (any experts around?). The Schroeder-Bernstein proof
> does not seem to generalize easily.
As I recall, it is not true. For the categry of groups, someone once
told me a counterexample, which I guess I could reconstruct given time.
For the categry of topological spaces, it is easily seen not to be
true (A = disk, B = disjoint union of 2 disks). For the category
of Banach spaces, I hear that it is not true, but this is non-trivial
(requires recent results of Gowers and Maurey).
--
Stephen Montgomery-Smith stephen@math.missouri.edu
307 Math Science Building stephen@showme.missouri.edu
Department of Mathematics stephen@missouri.edu
University of Missouri-Columbia
Columbia, MO 65211
USA
Phone (573) 882 4540
Fax (573) 882 1869
http://www.math.missouri.edu/~stephen