# ${C}^{n}$ norm

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

$${\parallel f\parallel}_{{C}^{n}}=\underset{x\in I}{sup}\underset{k\le n}{sup}\left|\frac{{d}^{k}f}{d{x}^{k}}\right|$$ |

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

$$\parallel \cdot {\parallel}_{{C}^{n}}$$ |

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^{}

$$ |

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,

$${\parallel {e}^{x}\parallel}_{{C}^{n}}=\mathrm{\infty}$$ |

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 “$\mathrm{max}$” 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 ${C}^{\mathrm{\infty}}$, there is no natural way to impose a norm, so instead one uses all the ${C}^{n}$ norms to define the topology^{} in ${C}^{\mathrm{\infty}}$. One does this by declaring that a subset of ${C}^{\mathrm{\infty}}$ 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 |
---|---|

Canonical name | CnNorm |

Date of creation | 2013-03-22 14:59:46 |

Last modified on | 2013-03-22 14:59:46 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 9 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 46G05 |

Classification | msc 26B05 |

Classification | msc 26Axx |

Classification | msc 26A24 |

Classification | msc 26A15 |