A sensible topology on the space of continuous linear maps between Fréchet spaces

Question

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?

Answer 1

For $V_2=\mathbb K$, the field of real or complex numbers, $L(V_1,V_2)=V_1'$ is the continuous dual of $V_1$. The generalization of the strong topology on $V_1'$ to $L(V_1,V_2)$ is the topology of uniform convergence on bounded sets which is described by the seminorms $p_{B,n}(T)=\sup\{\|T(x)\|_{2,n}: x\in B\}$ for $n\in\mathbb N$ and $B\subseteq V_1$ bounded. This makes $L(V_1,V_2)$ a complete locally convex space. Instead of all bounded sets one can consider smaller classes, e.g., uniform convergence on all finite or on all compact subsets of $V_1$.

None of these topologies makes $L(V_1,V_2)$ again a Fréchet space. However, this space of operators has the natural structure of countable projective limit of countable inductive limits of Banach spaces, a so-called PLB-space. This structure is used for splitting theorems for Fréchet spaces via the functor Ext$^1(V_1,V_2)$. If house advertisement is allowed, I would recommend my Springer Lecture Notes Derived Functors in Functional Analysis.