```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
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