# 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 ${\mathbb{R}}^{n}$ listed here as well as spaces of sequences and spaces of functions, and also complex ${L}^{p}$ spaces.

Title | Minkowski inequality |
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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 |