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

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?

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.
• Just one more. If $V_1$ and $V_2$ are Montel spaces (e.g. nuclear Frechet), then is $L(V_1,V_2)$ also a Montel space? You said it is a PLB-space, which involves Banach spaces. So, I am a bit confused here. Commented Jul 11 at 21:44
• At least, for nuclear Fréchet spaces, $L(V_1,V_2)$ is a PLB-space where the LB-spaces are (can be chosen to be) nuclear. Then $L(V_1,V_2)$ is also nuclear and hence Montel (this is somewhat implicitly alreqady contained in Grothendieck's thèse since $L(V_1,V_2)\cong V_1' \hat\otimes V_2$). Commented Jul 12 at 8:36
• @JochenWengenroth I am aware that a nuclear Frechet space is Montel, but you said $L(V_1,V_2)$ is not a Frechet space. I guess it is still Montel somehow? Commented Jul 12 at 13:10