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

Structure of $C^\infty(U)$ where $U\subseteq\mathbb{R}^n$ is open

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

• $C^\infty(U)$ is a vector space where:
  1. $(f+g)(x)=f(x)+g(x)$ (the addition operator) and
  2. $(\lambda f)(x) = \lambda f(x)$ (the scalar multiplication)
• $C^\infty(U)$ is an Algebra where:
  1. $(fg)(x)=f(x)g(x)$ is the product or multiplication operator