Question: Given a vector space $V$, is it possible to endow it with two different vector space topologies $\mathcal T_1$ and $\mathcal T_2$ such that any linear functional on $V$ is continuous in the sense of $\mathcal T_1$ if only if it is continuous in the sense of $\mathcal T_2$?
By a vector space topology I mean a topology such that vector addition and scalar multiplication are continuous.
I think it is quite a basic question, but I couldn't work out a answer.
Edit: Before I asked this question, I was actually thinking whether the the dual of $C_c^\infty(\mathbb R^n)$ remains the same when the space is given the usual topology of test functions and the subspace topology of $C_c(\mathbb R^n)$. Indeed I didn't think through the general question before I posted it on this site, and I didn't mean to do it that this question became a "hot network question".
This is a basic fact:
Let $V$ be a topological vector space (with topology $\tau_1$). Consider the weak topology $\tau_2$ on $V$ ( given by the duality $V\times V^{\star}\to k$). Then $V$ with $\tau_2$ has the same topological dual as $V$ with $\tau_1$ (easy to prove).
$\bf{Added:}$ Conclusion: for a topological vector space $(V, \tau_1)$, the weak topology on $V$ is the smallest topology having the same dual. I think there exists the notion of the strongest topology with the same dual. Maybe related to bornological spaces? I vaguely remember it. Reference: Schaeffer - Topological Vector spaces
$\bf{Added:}$ As @Jochen mentioned, the strongest topology is the Mackey topology .