- #1

Bacle

- 662

- 1

Just curious: Are any two infinite-dimensional vector spaces A,B over the

same field F isomorphic?

It is straightforward to show any two finite-dim. V.Spaces (over the

same field F, of course) are isomorphic. If V,W have dim. n< oo , and respective

bases {v1,..,vn}, {w1,...,wn} ; both V.Spaces

over F, then we can represent any v, any w in V,W respectively, by:

v=f1*v1+f2*v2+...+fn*vn

w=f1'*w1+f2'*w2+...+f'n*wn

where fi,fi' are in F; i in {1,2,..,n}.

And then the maps:

h: v-> (f1,..,fn) :V->F^n

h': w->(f1',..,fn'): W->F^n

are isomorphisms. Then the composition h'^-1o h

gives us an iso. between V,W.

Are A,B infinite-dimensional of the same cardinality (cardinality of dimension, of course)

isomorphic? Does the trick above also work for the infinite-dimensional case?

Thanks.