Sum and intersection of anihalator spaces

[talolard]

Homework Statement

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


First prove That
(U$$\bigcap$$W)$$^{\circ}$$$$\supseteq$$W$$^{\circ}$$+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$$\bigcap$$W)$$^{\circ}$$$$\supseteq$$W$$^{\circ}$$+U$$^{\circ}$$

Next prove
(U$$\bigcap$$W)$$^{\circ}$$$$\subseteq$$W$$^{\circ}$$+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 $$f \in (U \cap W)^\circ$$.