# Class of smooth real-valued functions on R-n

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:

• [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]

## 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] (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:
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