Date: Mar 5, 2013 5:44 PM
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