Thread: Annihilators in Banach Spaces

Let X be a normed vector space having Banach dual space X′, let L be a subset of X , let M be a subset on X ′
and let L◦up = {f ∈ X ′: f (x) = 0∀x ∈ L}, M◦down = {x ∈ X : f (x) = 0∀f ∈ M}, be the topological annihilators of L and M , respectively. Prove the following results:

(i) L◦ up and M◦down are closed subspaces of X′ and X , respectively.
(ii) Cl(linL)◦up = L◦up, Cl(linM )◦down = M◦ down .
(iii) Cl(linL) = (L◦up)◦down , Cl(linM) ⊆ (M◦down)◦up.

I can do the first part, but help with parts ii) and iii) would be very much appreciated.

(ii)Since Cl(lin L) contains L, thus L◦up contains Cl(linL)◦up .
And if f is in L◦up, then f lies in Cl(linL)◦up for f is linear and coutinous. Thus Cl(linL)◦up contains L◦up. Q.E.D for the first conclusion.
Similarly for the second conclusion!
(iii) can also be proved By using the definition and the linearity and continuousity of Bounded linear function.