Class of smooth real-valued functions on R-n

Note: the topology assumed on $\mathbb{R}^n$ here is the usual one, that is the one induced by the Euclidean norm

Definition

The class of all smooth, real-valued, functions on $\mathbb{R}^n$ is denoted:

• $C^\infty(\mathbb{R}^n)$

The conventions concerning the $C^k$ notation are addressed on the page: Classes of continuously differentiable functions This means that:

• $f\in C^\infty(\mathbb{R}^n)\iff[f:\mathbb{R}^n\rightarrow\mathbb{R}\wedge\ f\text{ is }$smooth$\text{ on }\mathbb{R}^n]$
  • Recall that to be smooth we require:

    $f$ be $k$-times continuously differentiable $\forall k\in\mathbb{Z}[k\ge 0]$

    Or indeed that: all partial derivatives of all orders exist and are continuous on $\mathbb{R}^n$

Generalising to open sets

Let $U\subset\mathbb{R}^n$ (for some $n$) be open in $\mathbb{R}^n$, then:

• $C^\infty(U)$

denotes the set of all functions, $:U\rightarrow\mathbb{R}$ that are smooth on $U$ (so all partial derivatives of all orders are continuous on $U$)

Structure

Let $U\subseteq\mathbb{R}^n$ be an open subset (notice it is non-proper, so $U=\mathbb{R}^n$ is allowed), then:

• $C^\infty(U)$ is a vector space where:
  1. $(f+g)(x)=f(x)+g(x)$ (the addition operator) and
  2. $(\lambda f)(x) = \lambda f(x)$ (the scalar multiplication)
• $C^\infty(U)$ is an Algebra where:
  1. $(fg)(x)=f(x)g(x)$ is the product or multiplication operator

See also

• Classes of continuously differentiable functions
• Derivation (Differential geometry)