fa.functional analysis - What functorial topologies are there on the space of linear maps between LCTVS?

If $V$ is a vector space, there is a least topology $tau_V$ on $V$ which makes all linear maps from $V$ to any other vector space continuous (consider the initial topology from the collection of all linear maps from $V$ to all vector spaces which are quotients of $V$---this is a set of maps, which is nice, but it is not really needed) This topology is locally convex: it is the largest locally convex topology, in fact.

If you endow $L(V,W)$, the space of continuous linear maps between two lcvts $V$ and $W$, with the topology $tau_{L(V,W)}$, then you get a functor.

(You can play this game in many ways: pick a set $mathcal F$ of your favorite locally convex topological vector spaces — for example, let $mathcal F={L^{2/3}(mathbb R), mathcal E'}$, — and endow $L(V,W)$, for all lctvs $V$ and $W$, with the least topology $tau_mathcal F$ for which all linear maps from $L(V,W)$ to an element of $mathcal F$ are continuous. This gives again a functor.) (Moreover, one can do the same with final topologies, of course)

Remark that these topologies are probably useless :)

Later: There is another source of examples, generalizing the uniform topologies.
Suppose you have an assignment to each lctvs $V$ of a set $C(V)subseteqmathcal P(V)$ of subsets of $V$ which is functorial, in the sense that whenever $f:Vto W$ is a continuous linear map of lctvs then ${f(A):Ain C(V)}subseteq C(W)$. Then there is functorial topology on $L(V,W)$ given by uniform convergence on sets of $C(V)$.

If you take $C(V)$ to be the set of bounded subsets of $V$, of compact subsets of $V$, of relatively compact subsets of $V$, of finite subsets of $V$, then you get your examples 1 to 4. But there are other choices for $C(V)$: the set of star shaped subsets of $V$ (ie, subsets $Ssubseteq V$ such that there is a point $xin S$ such that for all $yin S$ the segment $[x,y]$ is contained in $S$; this over $mathbb R$---over $mathbb C$ there is a corresponding notion whose name escapes me now), say. There are others.

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