# Concept derivation guides

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Concept guide



## [Norm-induced metric: proving from the norm axioms that d(x,y)=‖x−y‖ is a metric, with exact l1 and l∞ computations in ℝ³](/en/derivations/derivation-mko-norm-induced-metric-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-norm-induced-metric)

Let (V,‖·‖) be a real normed vector space. Prove that d(x,y)=‖x−y‖ is nonnegative, separates points, is symmetric and satisfies the triangle inequality; prove that the l1 and l∞ formulas on ℝⁿ are norms, so both induce metrics; then compute d1 and d∞ exactly in ℝ³ for x=(1,−2,3), y=(4,0,−1), z=(2,5,1).

Norm-induced metric

derivation-mko-norm-induced-metric-01





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