Minkowski inequality

If $p\ge 1$ and $a_k,b_k$ are real numbers for $k=1,\mathrm{\dots }$, then

$${\left(\sum _{k=1}^n{|a_k+b_k|}^p\right)}^{1/p}\le {\left(\sum _{k=1}^n{|a_k|}^p\right)}^{1/p}+{\left(\sum _{k=1}^n{|b_k|}^p\right)}^{1/p}$$

The Minkowski inequality is in fact valid for all $L^p$ norms with $p\ge 1$ on arbitrary measure spaces. This covers the case of ${ℝ}^n$ listed here as well as spaces of sequences and spaces of functions, and also complex $L^p$ spaces.

Title	Minkowski inequality
Canonical name	MinkowskiInequality
Date of creation	2013-03-22 11:46:24
Last modified on	2013-03-22 11:46:24
Owner	drini (3)
Last modified by	drini (3)
Numerical id	13
Author	drini (3)
Entry type	Theorem
Classification	msc 26D15
Related topic	LebesgueMeasure
Related topic	MeasurableSpace