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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 $L(C^\infty[a,b])$?

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The space $L(C^\infty[a,b])$ of continuous linear operators on $C^\infty[a,b]$ is - excepting the trivial case $a = b$ - not a Fréchet space (in the usual topologies).

The usual topologies on $L(C^\infty[a,b])$ are

  • the topology of uniform convergence on all bounded subsets, often (well, sometimes, at least ;) denoted $L_b(C^\infty[a,b])$,
  • the topology of uniform convergence on all compact subsets, often denoted $L_c(C^\infty[a,b])$,
  • and the topology of uniform convergence on all finite subsets (or pointwise convergence), also denoted by $L_p(C^\infty[a,b])$.

The seminorms generating the respective topology would be

$$p_{q,M}(T)=\sup\{q(Tx):x\in M\},$$

where $q$ is a continuous seminorm on $C^\infty[a,b]$, and $M$ ranges over the family of subsets of $C^\infty[a,b]$ under consideration, bounded, compact, or finite.

These topologies are not metrizable, since there is no countable basis of bounded/compact/finite sets in $C^\infty[a,b]$, hence no countable subset of seminorms generates the topology.

The space of linear differential operators (with smooth coefficients) is a subspace of $L(C^\infty[a,b])$, since such an operator is a continuous linear operator on $C^\infty[a,b]$.

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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.

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