Date: Mar 5, 2013 5:44 PM
Author: Kaba
Subject: Re: Orthogonal complement
5.3.2013 1:19, Kaba wrote:

>> My intuition says that these properties hold only for finite-dimensional

>> spaces, and therefore any proof must necessary use a property which is

>> specific to finite dimensional spaces.

Ok, I think I got this today. The sequence of Theorems is as follows:

1) A vector space V over F is isomorphic to finite(F^I), where I is any

set with |I| = dim(V), and finite(.) denotes that subset of F^I which is

non-zero only for finitely many i in I.

2) The dual space V* of V is isomorphic to F^I, where I is any set with

|I| = dim(V).

3) If F is a field, then F^I is isomorphic to finite(F^I) if and only if

I is a finite set.

It follows that

4) Vector space V is finite-dimensional if and only if V is isomorphic

to its dual space V*.

It is this theorem, together with the rank-nullity theorem, which opens

up the route to other finite-dimensional theorems, such as the Riesz

representation theorem for bilinear spaces, and then to those specific

orthogonal complement theorems.

I will comment on the other replies later, when I have gone through some

more proofs.

--

http://kaba.hilvi.org