Class of smooth real-valued functions on R-n
From Maths
- Note: the topology assumed on [ilmath]\mathbb{R}^n[/ilmath] here is the usual one, that is the one induced by the Euclidean norm
Definition
The class of all smooth, real-valued, functions on [ilmath]\mathbb{R}^n[/ilmath] is denoted^{[1]}:
- [ilmath]C^\infty(\mathbb{R}^n)[/ilmath]
The conventions concerning the [ilmath]C^k[/ilmath] notation are addressed on the page: Classes of continuously differentiable functions This means that:
- [ilmath]f\in C^\infty(\mathbb{R}^n)\iff[f:\mathbb{R}^n\rightarrow\mathbb{R}\wedge\ f\text{ is }[/ilmath]smooth[ilmath]\text{ on }\mathbb{R}^n][/ilmath]
- Recall that to be smooth we require:
- [ilmath]f[/ilmath] be [ilmath]k[/ilmath]-times continuously differentiable [ilmath]\forall k\in\mathbb{Z}[k\ge 0][/ilmath]
- Or indeed that: all partial derivatives of all orders exist and are continuous on [ilmath]\mathbb{R}^n[/ilmath]
- Recall that to be smooth we require:
Generalising to open sets
Let [ilmath]U\subset\mathbb{R}^n[/ilmath] (for some [ilmath]n[/ilmath]) be open in [ilmath]\mathbb{R}^n[/ilmath], then:
- [ilmath]C^\infty(U)[/ilmath]
denotes the set of all functions, [ilmath]:U\rightarrow\mathbb{R} [/ilmath] that are smooth on [ilmath]U[/ilmath]^{[1]} (so all partial derivatives of all orders are continuous on [ilmath]U[/ilmath])
Structure
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 where:
- [ilmath](fg)(x)=f(x)g(x)[/ilmath] is the product or multiplication operator
See also
References
- ↑ ^{1.0} ^{1.1} Introduction to Smooth Manifolds - John M. Lee - Second Edition