Example

— Let $\mathscr{C}^\infty(\mathbf{R})$ be the vector space over $\mathbf{R}$ of real valued functions that are infinitely differentiable in $\mathbf{R}$. For every function and every pair of integers $n \geq 0$, $m \geq 1$, put

with $f^{(0)} = f$. Obviously the $p_{n,m}$ are semi-norms on $\mathscr{C}^\infty(\mathbf{R})$. In order that the functions $f_\alpha$ tend to 0 (following a filter $\mathfrak{F}$ on the set of indices) in $\mathscr{C}^\infty(\mathbf{R})$ for the topology $\mathscr{T}$ defined by the semi-norms $p_{n,m}$, it is necessary and sufficient that for all integers $n \geq 0$, the functions $f_\alpha^{(n)}$ tend to 0 (following $\mathfrak{F}$) uniformly on every compact subset of $\mathbf{R}$. We say that $\mathscr{T}$ is the topology of compact convergence for the functions $f \in \mathscr{C}^\infty(\mathbf{R})$ and all their derivatives (cf. III, p. 9).