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# $C^{n}$ 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.

## Mathematics Subject Classification

46G05*no label found*26B05

*no label found*26Axx

*no label found*26A24

*no label found*26A15

*no label found*

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