PlanetMath: Minkowski inequality

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Minkowski inequality

(Theorem)

If $p \geq 1$ and are real numbers for $k = 1,\ldots$ , then

$\begin{displaymath}\left( \sum_{k=1}^n \vert a_k+b_k\vert^p\right)^{1/p} \le \le... ...right)^{1/p} + \left(\sum_{k=1}^n \vert b_k\vert^p\right)^{1/p}\end{displaymath}$

The Minkowski inequality is in fact valid for all norms with $p\ge1$ on arbitrary measure spaces. This covers the case of $\mathbbmss{R}^n$ listed here as well as spaces of sequences and spaces of functions, and also complex spaces.

"Minkowski inequality" is owned by drini. [ full author list (2) | owner history (2) ]

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See Also: Lebesgue measure, measurable space

Keywords:

measure

Attachments:: proof of Minkowski inequality (Proof) by Andrea Ambrosio

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Cross-references: complex, spaces of functions, sequences, covers, measure spaces, norms, real numbers
There are 3 references to this entry.

This is version 8 of Minkowski inequality, born on 2001-10-15, modified 2003-07-31.
Object id is 230, canonical name is MikowskiInequality.
Accessed 12221 times total.

Classification:

AMS MSC:

26D15 (Real functions :: Inequalities :: Inequalities for sums, series and integrals)

Pending Errata and Addenda

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