# Homework Help: Annihilator of annihilator

1. Oct 12, 2009

1. The problem statement, all variables and given/known data
Hi,

I have a problem where I have to show that any subset of S<V, where T:V->V**(dual dual space), identifies ann(ann(S)) and the span S.

2. Relevant equations

Given:
annS={f in V* | for any v in V T(v)=0 }
T(v)(f) = f(v)
T - linear isomorphism

3. The attempt at a solution

I think what the question asks is to show that T(spanS) = ann(ann(S))
Thus I need to show that v is in the spanS.

Given the def: annS={f in V* | for any v in V T(v)=0 } and the equality T(v)(f) = f(v), I wrote that ann(ann(S)) = {T(v) in V**| for any f in ann(S) Tv(f)=0 }

I started with
Tv=a_1v_1 +...+ a_nv_n
Tv(f) = a_1(f)v_1 + ...+ a_n(f)v_n = 0
Tv(f) = 0 for any f => Tv is in V**
Choose an arbitrary function f=X^2
then TV(f) = a_1*v_1^2 +...+ a_n*v_n^2
since sum of all v^2>0 then Tv(f) = 0 => a_1 ... a_n = 0
Thus v_1...v_n is in span S

this implies that T(spanS) = ann(ann(S))

For some reason I think it is wrong. I am not sure how to show the v is in the spanS

Thanks for any help!
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution