Annihilator of annihilator

[badsis]

Homework Statement

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.

Homework Equations

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

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!

 

[mighty2000]

Thank you for your question. It seems like you are on the right track with your solution. However, there are a few areas where you can improve your explanation. First, it would be helpful to define what V** and V* represent in the context of your problem. This will help the reader understand the notation and concepts involved.

Secondly, when showing that v is in the spanS, you can use the fact that T is a linear isomorphism to show that for any f in ann(S), there exists a unique v in V such that T(v) = f. This can be done by considering the definition of ann(S) and using the linearity of T.

Lastly, when choosing an arbitrary function f, it would be clearer to specify that f is a linear functional in V*, rather than just stating f=X^2. This will help to avoid any confusion or misinterpretation.

Overall, your solution is on the right track, but it would be helpful to provide more explanation and clarification in certain areas. Keep up the good work and don't hesitate to ask for further clarification if needed. Good luck with your studies!