It is standard that an $A$-module $P$ is projective iff. there exist elements $p_i \in P$ and $\bar{p}_i \in \hom_A(P, A)$ for $i$ in some indexing set $I$ such that for all $q \in P$, we have
$$q = \sum_{i \in I} \bar{p}_i(q) p_i$$
Does the opposite hold? I.e., is it true that under the same condition every $\lambda \in \hom_A(P, A)$ can be written as
$$\lambda = \sum_{i \in I} \lambda(p_i) \bar{p}_i ?$$
Likely, this cannot be true in general. Is it true if (and probably only if) $P$ is finitely generated?
Edit: I guess that if $P$ is finitely generated, then $P \oplus Q$ is free of finite rank for some $Q$ (although I'm not sure how to see that). In this case, it follows that $\hom_A(P, A)$ is also projective (by trading a finite product for a direct sum at some place) and thus has a dual basis. How can I see that I can use the same dual basis as for $P$?