===== Smooth function =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $X,Y$ ... Banach spaces with topology |
| @#FFBB00: definiendum | @#FFBB00: $f\in C^\infty(X,Y)$ |
| @#FFFDDD: for all     | @#FFFDDD: $k\in \mathbb N$ |
| @#55EE55: postulate   | @#55EE55: $f\in C^k(X,Y)$ |
| @#55EE55: postulate   | @#55EE55: $\lim_{m\to\infty}D^mf$ ... continuous |
==== Discussion ====
=== Formalities ===
Here $D^mf$ denotes the $m$th component of the sequence $Df, D^2f, D^3f,\dots$.
=== Reference ===
Wikipedia: [[http://en.wikipedia.org/wiki/Differentiable_function|Differentiable function]], [[http://en.wikipedia.org/wiki/Smooth_function|Smooth function]]
==== Parents ====
=== Context ===
[[Differentiable function]], [[Limit in a metric space]]