CnNorm 2005-02-02 12:45:14 2005-02-03 18:51:35 Definition $C^n$ % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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.