- #1
talolard
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Homework Statement
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