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

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:
  1. [ilmath](f+g)(x)=f(x)+g(x)[/ilmath] (the addition operator) and
  2. [ilmath](\lambda f)(x) = \lambda f(x)[/ilmath] (the scalar multiplication)
• [ilmath]C^\infty(U)[/ilmath] is an Algebra where:
  1. [ilmath](fg)(x)=f(x)g(x)[/ilmath] is the product or multiplication operator