| 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 |
Here $D^mf$ denotes the $m$th component of the sequence $Df, D^2f, D^3f,\dots$.
Wikipedia: Differentiable function, Smooth function