The most important metric spaces are normed vector spaces, which we want to introduce now.

Definition: Norm, Normed Vector Space

Let \(V\) be a vector space over the field of real numbers \(\mathbb R\). A norm is a map. \[\|~\|: \begin{cases} V&\mapsto \mathbb R\\ x&\mapsto \|x\|\\ \end{cases}\] with the following properties:

(1) \(\|x\|=0\) if and only if \(x=0\).

(2) \(\|\lambda x\|=|\lambda|\cdot\|x\|\) for all \(\lambda\in\mathbb R\).

(3) \(\|x+y\|\le \|x\|+\|y\|\) for all \(x,y\in V\).

If such a map exists, the pair \((V,\|~\|)\) is called a normed vector space.

  1. Proposition: p-Norm, Taxicab Norm, Euclidean Norm, Maximum Norm
  2. Proposition: Integral p-Norm

Definitions: 1 2 3 4 5 6 7 8
Parts: 9 10
Proofs: 11 12 13
Propositions: 14 15 16

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  1. Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984