Suppose x=(x1,,xn) is a point in n, and let xp and x be the usual p-norm and -norm;

xp = (|x1|p++|xn|p)1/p,
x = max{|x1|,,|xn|}.

Our claim is that

limpxp = x. (1)

In other words, for any fixed xn, the above limit holds. This, or course, justifies the notation for the -norm.

Proof. Since both norms stay invariant if we exchange two componentsPlanetmathPlanetmathPlanetmath in x, we can arrange things such that x=|x1|. Then for any real p>0, we have

x = |x1|=(|x1|p)1/pxp


xp n1/p|x1|=n1/px.

Taking the limit of the above inequalities (see this page ( we obtain

x limpxp,
limpxp x,

which combined yield the result.

Title limp\delimiter69645069x\delimiter86422285p=\delimiter69645069x\delimiter86422285
Canonical name limptoinftylVertXrVertplVertXrVertinfty
Date of creation 2013-03-22 14:02:53
Last modified on 2013-03-22 14:02:53
Owner Koro (127)
Last modified by Koro (127)
Numerical id 12
Author Koro (127)
Entry type Result
Classification msc 46B20
Related topic PowerMean