Cn norm

One can define an extended norm on the space Cn(I) where I is a subset of as follows:


If f is a function of more than one variable (i.e. lies in Cn(D) for a subset Dm), then one needs to take the supremum over all partial derivativesMathworldPlanetmath of order up to n.



satisfies the defining conditions for an extended norm follows trivially from the properties of the absolute valueMathworldPlanetmathPlanetmathPlanetmath (positivity, homogeneity, and the triangle inequalityMathworldMathworldPlanetmathPlanetmath) and the inequalityMathworldPlanetmath


If we are considering functions defined on the whole of m or an unboundedPlanetmathPlanetmath subset of m, the Cn norm may be infiniteMathworldPlanetmathPlanetmath. For example,


for all n because the n-th derivativePlanetmathPlanetmath of ex is again ex, which blows up as x approaches infinityMathworldPlanetmath. If we are considering functions on a compactPlanetmathPlanetmath (closed and boundedPlanetmathPlanetmathPlanetmathPlanetmath) subset of m however, the Cn norm is always finite as a consequence of the fact that every continuous functionMathworldPlanetmath 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 convergePlanetmathPlanetmath 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 Cn is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath under this norm. (In other words, it is a Banach spaceMathworldPlanetmath.)

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

Title Cn 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