Let $V_1$ and $V_2$ be Fréchet spaces. Let $\{ \lVert \cdot \rVert_{1,n} \}_{n \in \mathbb{N}}$ be a family of seminorms for $V_1$ and similarly $\{ \lVert \cdot \rVert_{2,n} \}_{n \in \mathbb{N}}$ for $V_2$.

Now, let $L(V_1, V_2)$ be the space of continuous linear maps from $V_1$ into $V_2$. Then, $L(V_1, V_2)$ is a vector space itself. However,

I cannot come up with a "sensible" (or "natural") topology for $L(V_1,V_2)$, making it a topological vector space.

If $V_1$ and $V_2$ are Banach spaces, then $L(V_1,V_2)$ itself is a Banach space with the operator norm.

I have checked if it is possible to apply the strong dual topology to $L(V_1,V_2)$. However, the strong dual topology requires a dual system to start with, so I don't think it is suitable for $L(V_1, V_2)$.

Could anyone help me?