On 04/03/2013 23:19, Kaba wrote: > 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.
This is true when V is finite-dimensional and S is a vector subspace of V.
The bilinear form induces a linear map g: V -> S* (S* being the dual space of S) which is surjective (by non-degeneracy of S, g is injective on S, so also surjective on S). By definition Ker g is C(S), so by rank-nullity dim C(S) = dim V - dim S* = dim V - dim S.
As g is surjective on S, for all u in V there is w in S with g(u) = g(w), that is u - w is in C(S). Therefore V = S + C(S) (dimension counting means the sum is direct).
Next we claim that C(S) is non-degenerate: any element of C(S) orthogonal to C(S) is also by definition orthogonal to S and so to V = S + C(S); thus it is zero.
Applying the above to C(S) gives dim C(C(S)) = dim(S). As S is clearly a subset of C(C(S)) then S = C(C(S)).