Smooth function

Set

 context $X,Y$ … Banach spaces with topology definiendum $f\in C^\infty(X,Y)$ for all $k\in \mathbb N$ postulate $f\in C^k(X,Y)$ postulate $\lim_{m\to\infty}D^mf$ … continuous

Discussion

Formalities

Here $D^mf$ denotes the $m$th component of the sequence $Df, D^2f, D^3f,\dots$.