One can define an extended norm on the space $C^n (I)$ where $I$ is a subset of $\mathbb{R}$ as follows: $$\| f \|_{C^n} = \sup_{x \in I} \sup_{k \le n} \left| \frac{d^k f}{dx^k} \right|$$ If $f$ is a function of more than one variable (i.e. lies in $C^n (D)$ for a subset $D \in \mathbb{R}^m$ ), then one needs to take the supremum over all partial derivatives of order up to $n$ .

That $$\| \cdot \|_{C^n}$$ satisfies the defining conditions for an extended norm follows trivially from the properties of the absolute value (positivity, homogeneity, and the triangle inequality) and the inequality $$\sup (|f| + |g|) < \sup |f| + \sup |g|.$$

If we are considering functions defined on the whole of $\mathbb{R}^m$ or an unbounded subset of $\mathbb{R}^m$ , the $C^n$ norm may be infinite. For example, $$\| e^x \|_{C^n} = \infty$$ for all $n$ because the $n$ -th derivative of $e^x$ is again $e^x$ , which blows up as $x$ approaches infinity. If we are considering functions on a compact (closed and bounded) subset of $\mathbb{R}^m$ however, the $C^n$ norm is always finite as a consequence of the fact that every continuous function on a compact set attains a maximum. This also means that we may replace the ``$\sup$ '' with a ``$\max$ '' in our definition in this case.

Having a sequence of functions converge under this norm is the same as having their $n$ -th derivatives converge uniformly. Therefore, it follows from the fact that the uniform limit of continuous functions is continuous that $C^n$ is complete under this norm. (In other words, it is a Banach space.)

In the case of $C^{\infty}$ , there is no natural way to impose a norm, so instead one uses all the $C^n$ norms to define the topology in $C^\infty$ . One does this by declaring that a subset of $C^\infty$ is closed if it is closed in all the $C^n$ norms. A space like this whose topology is defined by an infinite collection of norms is known as a multi-normed space.