Sum and intersection of anihalator spaces

[talolard]

Homework Statement

prove that (U\bigcapW)^{\circ}=W^{\circ}+U^{\circ}


First prove That
(U\bigcapW)^{\circ}\supseteqW^{\circ}+U^{\circ}

Take any f\in (U\bigcap W)^{\circ}
Then it is easy to see that for any f\in (U\bigcap W) f(v) =0

but since v\in U and v \in W then f \in U^{\circ} and f \in W^{\circ}



So we have

(U\bigcapW)^{\circ}\supseteqW^{\circ}+U^{\circ}

Next prove
(U\bigcapW)^{\circ}\subseteqW^{\circ}+U^{\circ}

W^{\circ} + U ^{\circ} = span(W)^{\circ} + span(U)^{\circ}

because

S^{\circ} =Span(S)^{\circ}

span(W)^{\circ} + span(U)^{\circ} = span (span(W)^{\circ} \cup span(u)^{\circ}

By definition of addition of subspaces

span (span(W)^{\circ} \cup span(u)^{\circ}= span (W \cup U) ^{\circ}

Which I am not sure of

And after all of that, we know that span (W \cap U) ^{\circ} \subseteq span (W \cup U) ^{\circ}

Which proves it if I did not make a mistake? Am I correct?

Thanks
Tal

 

[ystael]

Need some more information here. What are U and W?

 

[talolard]

Sorry Finite subspaces of the finite space V

 

[ystael]

Sorry Finite subspaces of the finite space V

I'll assume you mean "finite-dimensional".

In the first half, your argument is both faulty and goes in the wrong direction. You say you intend to prove (U \cap W)^\circ \supset U^\circ + W^\circ, and then you give an argument that begins with "take f \in (U \cap W)^\circ" and concludes that "f \in U^\circ + W^\circ". This argument, if correct, would prove (U \cap W)^\circ \subset U^\circ + W^\circ, not \supset : A \subset B means that \alpha \in A implies \alpha \in B.

However, the argument itself is not correct. If f \in (U \cap W)^\circ, and v \in U \cap W, then you are correct that f(v) = 0. However, you cannot conclude from this that f \in U^\circ on the grounds that f(v) = 0 and v \in U. To conclude that f \in U^\circ you would have to prove that f(v) = 0 for every v \in U, and this need not be true. What you actually want to do is give an equation f = g + h where g \in U^\circ and h \in W^\circ; this proves that f \in U^\circ + W^\circ.

In the second half, I can't understand at all what you've written. This direction, U^\circ + W^\circ \subset (U \cap W)^\circ, actually has a simpler, direct argument like the above: take f \in U^\circ + W^\circ, and prove that f \in (U \cap W)^\circ.