Annihilators

If V is a vector space over the field F and S is a subset of V,the annihilator of S is the set S^o of all linear functional f on V such that

f(α)=0  ⩝ α∊S

Sometimes A(S) is also used to denote the annihilators of S.

Thus            S⁰={f∊V’: f(α)=0 ⩝ α∊S}

Annihilator of an anhilator:-

Let V be a vector space over the field F.If S is any subset of V,then S⁰ is a subspace of V’.By definition of an annihilator, we have

(S⁰)⁰=S⁰⁰={L∊V’: L(f)=0 ⩝ f∊S⁰}

Obviously S⁰⁰ is a subspace of V’’.But if V is finite dimensional then we have identified V’’ with V through the natural isomorphism 𝛼⟺L_𝛼. Therefore we may regard S⁰⁰ as a subspace of V.Thus

S⁰⁰={α∊V : f(α)=0 ⩝ f∊S⁰}