Topology on set of smooth functions $\mathbb{R} \supset I \to \mathbb{R}$

Question

Let $I$ be a (connected) interval of the real line. I am interested in the set $X$ of all smooth functions $I \to \mathbb{R}$.

Does $X$ have a natural choice of topology?

Answer 1

The topology of uniform convergence of all derivatives on $X=C^\infty(I)$ is given by the sequence of seminorms $\|f\|_n=\sup\{|f^{(k)}(x)|: x\in I_n, 0\le k\le n\}$ where $I_n$ is an increasing sequence of compact intervalls with union $I$. Some reasons why this should be the most natural choice: It makes the algebraic operations on $C^\infty(I)$ (sums, products, derivatives) continuous, it is metrizable so that you can check continuity and other topological questions with sequences. For all analytical questions it is crucial that the space is complete. Moreover, $C^\infty(I)$ is a Montel space, i.e., the bounded sets are relatively compact which can be very helpful for existence proofs (it is even better than just Montel, a so-called Schwartz space). Finally, it is a nuclear Frechet space which very convenient e.g. when you deal with vector valued smooth functions.