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.
>



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