Let $V$ be a topological $k$-vector space.

Let $V^{\star}$ denote the space of all linear functionals $V \rightarrow k$ and $V' \subset V^{\star}$ the subspace of all continuous linear functionals.

The weak (Mackey) topology is the weakest (strongest) topology on the vector space $V$, such that all continuous functionals $v' \in V'$ remain continuous, and all discontinuous functionals $v' \in V^{\star} \setminus V'$ remain discontinuous w.r.t. the new weak (Mackey) topology.

Let $V_{\sigma} (V_{\tau})$ denote the underlying vector space $V$ equipped with its weak (Mackey) topology. In my opinion its clear that the weak topology on $V$ is an initial topology w.r.t. the family V', i.e. a function $f: U \rightarrow V_{\sigma}$ is continuous iff $v' \circ f$ is continuous for all $v' \in V'$.

By construction, if $f: U \rightarrow V_{\tau}$ is continuous, also $v' \circ f$ is continuous for all $v' \in V'$.

Does the inverse also hold? I.e. is the Mackey topology also initial? (No its not final)

Thank you.