# $C^{n}$ norm

One can define an extended norm on the space $C^{n}(I)$ where $I$ is a subset of $\mathbb{R}$ as follows:

 $\|f\|_{C^{n}}=\sup_{x\in I}\sup_{k\leq n}\left|\frac{d^{k}f}{dx^{k}}\right|$

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.

Title $C^{n}$ norm CnNorm 2013-03-22 14:59:46 2013-03-22 14:59:46 rspuzio (6075) rspuzio (6075) 9 rspuzio (6075) Definition msc 46G05 msc 26B05 msc 26Axx msc 26A24 msc 26A15