Let $E$ be a locally convex (Hausdorff) topological vector space. It's known that if $G$ is a linear subspace of $E$, then if $E$ has weak topology, then $G$ as a subspace also has weak topology.
We can define the Mackey topology on $E$ as the topology of uniform convergence on all of balanced convex $w^*$-compact subsets of $E^*$. Equivalently, it's the largest locally convex topology for which the continuous dual is $E^*$.
If we equip $E$ with Mackey topology, then does $G$ as a subspace also has Mackey topology? What if $G$ is closed?
Here's a result that might be helpful: If $q:E^*\to E^*/G^\perp$ is quotient map, and the topology on $E$ is that of uniform convergence on family $\mathcal{M}$, then the topology on $G$ as a subspace is that of uniform convergence on $$\widehat{\mathcal{M}} = \{q(M): M\in \mathcal{M}\}$$ where $G^*$ and $E^*/G^\perp$ are naturally identified with each other.
If $E$ is a Banach space then norm and Mackey topologies coincide. So the result is true for closed subspaces of Banach spaces.