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Mathematik-Online problems:

Problem 457: P-Norm

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
The set $ \left\{ x \in \mathbb{R}^n: \vert x_1\vert^p+ \cdots + \vert x_n\vert^p \leq 1 \right\}$ is convex for $ 1 \leq p \le \infty$. Use this relation to show that

$\displaystyle \Vert x \Vert _p := \left( \vert x_1\vert^p + \cdots + \vert x_n\vert^p\right)^{1/p} $

is a norm. Also proof the inequality $ \Vert x \Vert _p \leq \Vert x \Vert _q $, $ p \geq q$, and show, that $ \lim\limits_{p\rightarrow \infty} \Vert x \Vert _p = \max \vert x_i\vert$ holds.
(Authors: Höllig/Höfert)

see also:

  automatisch erstellt am 18.  1. 2017