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 $Df, D^2f, D^3f,\dots$.

Reference

Wikipedia: Differentiable function, Smooth function

Parents

Context

Differentiable function, Limit in a metric space