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

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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