I’m studying dual spaces in linear algebra.
I have proved the following two results. Note: I say $U^0$ for the annihilation of $U$.
For subspaces $U,W$ of a vector space $V$,
$(U+W)^0=U^0\cap W^0$
And
$U^0+W^0\subseteq (U\cap W)^0$
With the inequality becoming equality when $V$ is finite-dimensional.
I’m struggling to see the intuition for these results/visualise them/remember them.
Is there a good way to remember these results? Or sense check any other rearranging of these symbols would be false?
Thanks