Here's the problem I've been trying to get my mind around:
Prove that there exists an isomorphism of [tex]F^n[/tex] into [tex]Hom(Hom(F^n,F),F)[/tex].
I'm missing something. Here's what I get to:
[tex]F^n[/tex] is an n-tuple and [tex] F[/tex] is a field. So I can see that there is a set of homomorphisms from [tex]F^n[/tex] into [tex]F[/tex].
It would be a finite n-tuple mapped into an infinite field, so there would just be a finite number of elements mapped infinitely.
I used n=3 and the real numbers as an example for the sake of trying to understand this:
Define a homomorphism [tex]T:F^n \rightarrow F[/tex] as follows:
so [tex]x_3[/tex] is everything else besides 1 and 2.
Then I get confused trying to define [tex]Hom(Hom(F^n,F),F)[/tex] Am I just mapping everything back into the origional [tex]F[/tex]? And isn't that just what I started with, which is [tex]F[/tex]?(or the reals in my attempted example)
So the whole point is to show that there is an isomorphim from [tex]F^n[/tex] into the [tex]Hom((HomF^n,F),F)[/tex] But it looks to me like F^n is a finite n-tuple and I can't get my mind around how there can be an isomprphism between an infinite field and a finite n-tuple.
What am I missing? Where have I gone wrong?
any clarification will be greatly appreciated.
The Attempt at a Solution