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