Proof
(related to Proposition: Distance in Normed Vector Spaces)
The properties of a distance follow immediately from the definition of the norm:
- \(\|x-y\|=0\) if and only if \(x=y\).
- Symmetry: \(\|x-y\|=\|-1(y-x)\|=|-1|\cdot\|y-x\|=\|y-x\|\) for all \(x,y\in (V,\|~\|)\).
- Symmetry: \(\|x-y\|=\|-1(y-x)\|=|-1|\cdot\|y-x\|=\|y-x\|\) for all \(x,y\in (V,\|~\|)\).
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References
Bibliography
- Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984
