Let $V$ be a vector space, and suppose $V = U \oplus W$. Prove that $V^* = A(U) \oplus A(W)$.
In the finite dimensional case this is easy: I have the two results $A(U+W) = A(U) \cap A(W)$ and $A(U \cap W) = A(U) + A(W)$. Unforunately in the infinte dimensional case, I only have the first equality and the inclusion $A(U) + A(W) \subset A(U \cap W)$ and cannot proceed.