Denote by $\mathscr{D}^\prime$ and $\mathscr{D}^\prime_b$ the space of distributions on $\mathbb{R}^n$ equipped with the weak and the strong topology, respectively. Because the topology of $\mathscr{D}^\prime_b$ has more open sets than the topology of $\mathscr{D}^\prime$, we have $$L(\mathscr{D}^\prime, F) \subseteq L(\mathscr{D}^\prime_b, F)$$ for any topological vector space $F$. If $F = \mathbb{C}$, then we have equality (this holds for every locally convex space instead of $\mathscr{D}$). If $F = \mathscr{D}^\prime_b$, the left-hand-side is a proper subset because the identity is not continuous from $\mathscr{D}^\prime$ to $\mathscr{D}^\prime_b$ as the latter topology contains strictly more open sets (but the identity is continuous from each topological space to itself).

**Question:** What if $F = \mathscr{D}$, the space of test functions or $F = \mathscr{D}^\prime$, the space of distributions with the weak topology? Is this a strict inclusion or do we happen to have inequality?

If we don't have equality, what would be an example of a linear map that is contained in the latter space, but not in the first?

I didn't specify the domain yet. Is the answer different when we take the domain of functions in $\mathscr{D}$ to be a compact subset of $\mathbb{R}^n$ or a compact manifold instead of $\mathbb{R}^n$? What if we change the space $\mathscr{D}$ to the spaces $\mathscr{S}$ of Schwartz functions or $\mathscr{E}$ of all smooth functions, with there usual (Fréchet) topologies?