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prove that (U[tex]\bigcap[/tex]W)[tex]^{\circ}[/tex]=W[tex]^{\circ}[/tex]+U[tex]^{\circ}[/tex]

First prove That

(U[tex]\bigcap[/tex]W)[tex]^{\circ}[/tex][tex]\supseteq[/tex]W[tex]^{\circ}[/tex]+U[tex]^{\circ}[/tex]

Take any [tex]f\in (U\bigcap W)^{\circ}[/tex]

Then it is easy to see that for any [tex] f\in (U\bigcap W) f(v) =0 [/tex]

but since [tex] v\in U and v \in W [/tex] then [tex] f \in U^{\circ} and f \in W^{\circ}[/tex]

So we have

(U[tex]\bigcap[/tex]W)[tex]^{\circ}[/tex][tex]\supseteq[/tex]W[tex]^{\circ}[/tex]+U[tex]^{\circ}[/tex]

Next prove

(U[tex]\bigcap[/tex]W)[tex]^{\circ}[/tex][tex]\subseteq[/tex]W[tex]^{\circ}[/tex]+U[tex]^{\circ}[/tex]

[tex] W^{\circ} + U ^{\circ} = span(W)^{\circ} + span(U)^{\circ} [/tex]

because

[tex] S^{\circ} =Span(S)^{\circ} [/tex]

[tex] span(W)^{\circ} + span(U)^{\circ} = span (span(W)^{\circ} \cup span(u)^{\circ} [/tex]

By definition of addition of subspaces

[tex] span (span(W)^{\circ} \cup span(u)^{\circ}= span (W \cup U) ^{\circ} [/tex]

Which I am not sure of

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

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

Thanks

Tal

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# Homework Help: Sum and intersection of anihalator spaces

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