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

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==Structure of {{M|C^\infty(U)}} where {{M|U\subseteq\mathbb{R}^n}} is open==
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Let {{M|U\subseteq\mathbb{R}^n}} be an open subset (notice it is ''non-proper'', so {{M|1=U=\mathbb{R}^n}} is allowed), then:
* {{M|C^\infty(U)}} is a [[vector space]] where:
*# {{M|1=(f+g)(x)=f(x)+g(x)}} (the ''addition'' operator) and
*# {{M|1=(\lambda f)(x) = \lambda f(x)}} (the ''scalar multiplication'')
* {{M|C^\infty(U)}} is an [[Algebra]] where:
*# {{M|1=(fg)(x)=f(x)g(x)}} is the ''product'' or ''multiplication'' operator
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{{Definition|Manifolds|Differential Geometry|Real Analysis|Measure Theory}}
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