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Proposition: Distance in Normed Vector Spaces

Let \((V,\|~\|)\) be a normed vector space. Then \[d(x,y):=\|x-y\|,~~~~~~~(x,y\in V)\] defines a distance on \(V\)1.

Table of Contents

Proofs: 1

Mentioned in:

Definitions: 1
Proofs: 2 3

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References

Bibliography

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

Footnotes

  1. We also may use the notation \(\|x,y\|\) instead of \(\|x-y\|\), since \(\|x-y\|=\|y-x\|\) and it does not matter if we express the distance as \(\|x-y\|\) or \(\|y-x\|\). ↩