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Prove that $\operatorname{Ann}(U \cap W) = \operatorname{Ann}(U) + \operatorname{Ann}(W)$ if $\dim V < \infty$ for $U$ and $W$ subspaces of $V$.

Annihilator of U = Ann($U$) = $\{ \phi \in V^* | \phi(u) = 0 \text{ for } u \in U\}$.

Here $V^*$ is the dual space of $V$.

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What is $\mathrm{Ann}(U)$? is it the linear forms that vanish on $U$? –  Olivier Bégassat Feb 3 '13 at 23:42
    
Sorry, no. The annihilator of $U$. I'll add an edit. –  user4593 Feb 4 '13 at 0:00
    
What is $\operatorname{Ann}(U)$? Should that be $\dim U$ or dimension of $V/U$. Are you in an inner product space over $\mathbb R$ and speaking of the dimension of the orthogonal complement? $\operatorname{Ann}$ usually means something somewhere is multiplying something to zero. Please edit the question to make it intelligible. –  Barbara Osofsky Feb 4 '13 at 0:13
    
@BarbaraOsofsky his question makes sense to me. He defined Ann in the question, and I thought this was standard terminology although ususally denoted $U^0$, not $\operatorname{Ann}(U)$. I'm not entirely sure what your problem with his dimension thing is, how could this mean anything other than that $V$ has a finite dimension? –  Sam DeHority Feb 4 '13 at 0:30
    
@BarbaraOsofsky I think if you read his post again you'll find that all the necessary information is already there. –  Olivier Bégassat Feb 4 '13 at 0:30
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2 Answers

Here's a proof. Since linear forms that vanish on all of $U$ certainly vanish on all of $U\cap W$, one has $\mathrm{Ann}(U\cap W)\supset \mathrm{Ann}(U)$. Similarly $\mathrm{Ann}(U\cap W)\supset \mathrm{Ann}(W)$, so $$\mathrm{Ann}(U\cap W)\supset \mathrm{Ann}(U)+\mathrm{Ann}(W)$$ We dualize this to get an inclusion in the double dual space: $$\mathrm{Ann}\Big(\mathrm{Ann}(U\cap W)\Big)\subset \mathrm{Ann}\Big(\mathrm{Ann}(U)+\mathrm{Ann}(W)\Big)$$


On the other hand, if a linear form vanishes on all of $U + W$ , then it surely vanishes on both $U$ and $W$, so $$\mathrm{Ann}(U+ W)\subset \mathrm{Ann}(U)\cap\mathrm{Ann}(W)$$ Use this second fact with the subspaces $U'=\mathrm{Ann}(U)\subset V^*$ and $W'=\mathrm{Ann}(W)\subset V^*$. This gives you $$\mathrm{Ann}\Big(\mathrm{Ann}(U)+ \mathrm{Ann}(W)\Big)\subset \mathrm{Ann}\Big(\mathrm{Ann}(U)\Big)\cap\mathrm{Ann}\Big(\mathrm{Ann}(W)\Big)$$


Combine the two facts and you get the inclusions $$\mathrm{Ann}\Big(\mathrm{Ann}(U\cap W)\Big)\subset \mathrm{Ann}\Big(\mathrm{Ann}(U)+\mathrm{Ann}(W)\Big)\subset \mathrm{Ann}\Big(\mathrm{Ann}(U)\Big)\cap\mathrm{Ann}\Big(\mathrm{Ann}(W)\Big)$$


Under the canoncal isomorphism $\varphi:V\simeq V^{**},~x\mapsto\text{ (evaluation at }x)$ , this is where the finite dimension of $V$ comes into play, one has $\mathrm{Ann}(\mathrm{Ann}(U))=\varphi(U)$. Apply $\varphi^{-1}$ to the term on the left and the one on the right, and you get that they are in fact the same, so $$\mathrm{Ann}\Big(\mathrm{Ann}(U\cap W)\Big)= \mathrm{Ann}\Big(\mathrm{Ann}(U)+\mathrm{Ann}(W)\Big)=\mathrm{Ann}\Big(\mathrm{Ann}(U)\Big)\cap\mathrm{Ann}\Big(\mathrm{Ann}(W)\Big)$$


As a last step, dualize one more time: $$\mathrm{Ann}\bigg( \mathrm{Ann}\Big(\mathrm{Ann}(U\cap W)\Big)\bigg) = \mathrm{Ann} \bigg(\mathrm{Ann}\Big(\mathrm{Ann}(U)+\mathrm{Ann}(W)\Big)\bigg)$$ using the canonical isomorphism between $V^*$ and $(V^*)^{**}$, we get $$\mathrm{Ann}(U\cap W)=\mathrm{Ann}(U)+\mathrm{Ann}(W).$$

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It's pretty clear that some functional $\Lambda$ annihilates $U+V$ iff it annihilates $U$ and $V$. So we have $$\operatorname{Ann}(U+W) = \operatorname{Ann}(U) \cap \operatorname{Ann}(W)$$ You didn't want this, but it's an easy and related result. If $\Lambda = u+w \in \operatorname{Ann}(U)+\operatorname{Ann}(W)$, with $u$ coming from $\operatorname{Ann}(U)$ and $w$ from $\operatorname{Ann}(W)$, then both $s$ and $t$, and therefore $\Lambda$ are in $\operatorname{Ann}(U\cap W)$. We therefore have the inclusion: $$\operatorname{Ann}(U)+\operatorname{Ann}(W) \subseteq \operatorname{Ann}(U\cap W)$$

We now need to prove inclusion the the reverse direction, so we show that an element $\Lambda \in \operatorname{Ann}(U\cap W)$ is in $\operatorname{Ann}(U)+\operatorname{Ann}(W)$. We do this as follows. If $U = U'\oplus(U \cap W),W= (U \cap W)\oplus W'$ we have $$V = U' \oplus (U \cap V) \oplus W' \oplus Q$$ for some $Q$. We then define two functionals $\sigma,\tau \in V^*$. $$\sigma|U' = \Lambda,~~~~~ \sigma|(U\cap W) = 0~~~~~ \sigma|W' = 0, ~~~~~ \sigma|Q = \Lambda\\ \tau|U' = 0,~~~~~ \tau|(U\cap W) = 0~~~~~ \tau|W' = \Lambda, ~~~~~ \tau|Q = 0$$ And it is easy to see that $\sigma \in \operatorname{Ann}(U), \tau \in \operatorname{Ann}(W)$, as well as that $\sigma + \tau = \Lambda$, which gives the other inclusion $$\operatorname{Ann}(U)+\operatorname{Ann}(W) \supseteq \operatorname{Ann}(U\cap W)$$ and proves the theorem.

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