TITLE: Linear operators on $C^\infty[a,b]$
QUESTION [6 upvotes]: I do not know too much about linear operators so forgive me if this doesn't make much sense, but what would the space of linear operators on $C^\infty[a,b]$ be defined as? 
If we denote this space as $L(C^\infty[a,b])$ what can we say about it? Since $C^\infty[a,b]$ is a Frechet space is $L(C^\infty[a,b])$ also a Frechet space? What would be a useful norm (seminorms?) on $L(C^\infty[a,b])$? Given a norm or seminorms could you then talk about sequences of operators and their convergence? 
Also given that the space $L(C^\infty[a,b])$ exists would the space of linear differential operators on $C^\infty[a,b]$ be a subspace of $L(C^\infty[a,b])$?

REPLY [4 votes]: Adding a little to @DanielFischer's answer: the space $C^\infty[a,b]$, being a nested intersection, is a projective limit of Banach spaces. Thus, any (continuous linear) map to it from a TVS is exactly induced (uniquely) from a collection of (continuous linear) maps to the limitands $C^k[a,b]$. In particular, this is true of self-maps, so every such just comes from a compatible family of maps $C^\infty[a,b]\to C^k[a,b]$. Since the latter is Banach, it is an easy exercise to see that such a map factors through some $C^\ell[a,b]\to C^k[a,b]$ (with $\ell$ depending on $k$). Maps between Banach spaces have a natural selection of topologies on them, for example the operator-norm, but also the strong and weak, as delineated by DanielFischer.
Further, possibly relevantly depending on one's situation, there is a Schwartz kernel theorem valid for maps $C^\infty[a,b]\to (C^\infty[a,b])^*$ from smooth functions to distributions... namely, that every such is given by a distribution in two variables.