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Topic: Orthogonal complement
Replies: 15   Last Post: Mar 5, 2013 6:48 PM

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 Kaba Posts: 289 Registered: 5/23/11
Re: Orthogonal complement
Posted: Mar 5, 2013 5:44 PM

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