Class of smooth real-valued functions on R-n/Structure
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Revision as of 23:42, 21 October 2015 by Alec (Talk | contribs) (Created page with "<noinclude> ==Structure of {{M|C^\infty(U)}} where {{M|U\subseteq\mathbb{R}^n}} is open== </noinclude> Let {{M|U\subseteq\mathbb{R}^n}} be an open subset (notice it is ''non-p...")
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]C^\infty(U)[/ilmath] is an Algebra
