# Class of smooth real-valued functions on R-n/Structure

From Maths

## Structure of [ilmath]C^\infty(U)[/ilmath] where [ilmath]U\subseteq\mathbb{R}^n[/ilmath] is open

Let [ilmath]U\subseteq\mathbb{R}^n[/ilmath] be an open subset (notice it is *non-proper*, so [ilmath]U=\mathbb{R}^n[/ilmath] is allowed), then:

- [ilmath]C^\infty(U)[/ilmath] is a vector space where:
- [ilmath](f+g)(x)=f(x)+g(x)[/ilmath] (the
*addition*operator) and - [ilmath](\lambda f)(x) = \lambda f(x)[/ilmath] (the
*scalar multiplication*)

- [ilmath](f+g)(x)=f(x)+g(x)[/ilmath] (the
- [ilmath]C^\infty(U)[/ilmath] is an Algebra where:
- [ilmath](fg)(x)=f(x)g(x)[/ilmath] is the
*product*or*multiplication*operator

- [ilmath](fg)(x)=f(x)g(x)[/ilmath] is the