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

: '''Note: ''' the topology assumed on {{M|\mathbb{R}^n}} here is the usual one, that is the one induced by the [[Euclidean norm]]
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==Definition==
The ''class of all [[smooth]], [[real-valued function|real-valued]], [[function|functions]] on {{M|\mathbb{R}^n}} is denoted<ref name="ITSM">Introduction to Smooth Manifolds - John M. Lee - Second Edition</ref>:
* {{M|C^\infty(\mathbb{R}^n)}}
{{Note|The conventions concerning the {{M|C^k}} notation are addressed on the page: ''[[Classes of continuously differentiable functions]]''}}
This means that:
* {{M|1=f\in C^\infty(\mathbb{R}^n)\iff[f:\mathbb{R}^n\rightarrow\mathbb{R}\wedge\ f\text{ is } }}[[smooth]]{{M|\text{ on }\mathbb{R}^n]}}
** Recall that to be smooth we require:
**: {{M|f}} be [[k-times continuously differentiable|{{M|k}}-times continuously differentiable]] {{M|\forall k\in\mathbb{Z}[k\ge 0]}}
**: Or indeed that: all partial derivatives of all orders exist and are continuous on {{M|\mathbb{R}^n}}
==Generalising to open sets==
Let {{M|U\subset\mathbb{R}^n}} (for some {{M|n}}) be open in {{M|\mathbb{R}^n}}, then:
* {{M|C^\infty(U)}}
denotes the set of all functions, {{M|:U\rightarrow\mathbb{R} }} that are [[smooth]] on {{M|U}}<ref name="ITSM"/> (so all partial derivatives of all orders are continuous on {{M|U}})
==[[Class of smooth real-valued functions on R-n/Structure|Structure]]==
{{:Class of smooth real-valued functions on R-n/Structure}}
==See also==
* [[Classes of continuously differentiable functions]]
* [[Derivation (Differential geometry)]]
==References==
<references/>
{{Definition|Manifolds|Functional Analysis|Real Analysis|Measure Theory}}
[[Category:Examples of algebras]]