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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?

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1 Answer 1

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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.

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    $\begingroup$ I won't edit in a link in case you prefer not to link to your own work, but here it is in a comment: Wengenroth - Derived functors in functional analysis. $\endgroup$
    – LSpice
    Commented Jul 9 at 14:44
  • $\begingroup$ Thank you for your clarification and reference! $\endgroup$
    – Isaac
    Commented Jul 9 at 14:55
  • $\begingroup$ 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. $\endgroup$
    – Isaac
    Commented Jul 11 at 21:44
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    $\begingroup$ 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 $L(V_1,V_2)\cong V_1' \hat\otimes V_2$). $\endgroup$ Commented Jul 12 at 8:36
  • $\begingroup$ @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? $\endgroup$
    – Isaac
    Commented Jul 12 at 13:10

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