Date: Mar 4, 2013 6:19 PM
Author: Kaba
Subject: Re: Orthogonal complement
5.3.2013 1:17, Kaba kirjoitti:

> Claim

> -----

>

> If S subset V is a non-degenerate subspace of V, then C(C(S)) = S.

This should have read:

If S subset V, where V is a non-degenerate symmetric bilinear space,

then C(C(S)) = S.

> Attempted proof sketch

> ----------------------

>

> First it is shown that S subset C(C(S)), which is easy to see. Then,

> without additional justifications, it is claimed that

>

> dim(S) + dim(C(S)) = dim(V) = dim(C(S)) + dim(C(C(S))),

>

> which implies dim(C(C(S)) = dim(S), and therefore C(C(S)) = S.

>

> Related

> -------

>

> There is a similar argument in Lang's Algebra, equally mysterious. What

> might be the source of such dimension arguments? I'm guessing the

> rank-nullity theorem:

>

> dim(V) = dim(f(V)) + dim(f^{-1}(0)),

>

> for any linear f : V --> W.

>

> The problem is that while it is possible to prove, for V non-degenerate,

> that

>

> S intersect C(S) = {0},

>

> it seems hard to prove that S union C(S) = V. In general, these

> questions seem related:

>

> 1) C(C(S)) = S

> 2) V = S + C(S) (direct sum)

> 3) dim(S) + dim(C(S)) = dim(V)

>

> 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.

>

--

http://kaba.hilvi.org