$W_1$ and $W_2$ are subspaces of a finite-dimensional vector space $V$. $W^0$ is the annihilator of $W$.
(a) Prove $(W_1 + W_2)^0 = W_1^0 \cap W_2^0$.
(b) Prove $(W_1 \cap W_2)^0 = W_1^0 + W_2^0$.
Attempt :
\begin{align} (a)\,f\in (W_1+W_2)^0 &\iff f\in \{g\in V^*\mid g(\alpha_1 +\alpha_2)=0\, \forall \alpha_1 \in W_1, \alpha_2 \in W\} \\ & \iff f\in \{g\in V^*\mid g(\alpha_1)=0 \,\&\,g(\alpha_2)=0\,\forall \alpha_1\in W_1,\alpha_2 \in W_2\}\\ & \iff f\in W_1^0\cap W_2^0\\ (b)\,f\in (W_1^0+W_2^0) &\iff f\in \{g\in V^*\mid g=f_1+f_2\,\forall f_1\in W_1^0,f_2\in W_2^0\}\\ &\iff f\in \{g\in V^*\mid g(\alpha)=0,\alpha\in W_1\cap W_2\}\\ &\iff f\in (W_1\cap W_2)^0 \end{align}
This question has been asked before. But I wanted the proof to look short by just using the if and only if statements. Does my argument look right? Are there any steps that need to be inserted in between so that the proof finds some clarity.
