null-space and annihilators

I tried to work the following two rules, if someone can please confirm to me that they are indeed correct and guide how can I prove/counter-prove them

V - n dim vector space

V* dual space of V i.e. Hom(V,F)

Let S be a sub-set of V. A(S) is annihilator of S (thus is a sub-space of V*)

Let F be a sub-set of V*. N(F) is null-space of F (which means all v in V, such that for all f in F f(v)=0). N(F) can be shown to be a sub-space of V.

L(X) is linear span of any sub-set of any vector space.

Under the above notations, I want to establish/dis-prove the following:

1. N(A(S)) = L(S)

2. A(N(F)) = L(F)

Will really appreciate help here. I am just getting too confused here.