# Norm-induced metric

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

On the same real normed vector space, taking the norm of a vector difference defines a metric whose axioms follow universally from the norm axioms.

mko-norm-induced-metric · definition · v0.10.0



## Mathematical statement

If (V,norm) is a certified real normed vector space, d(x,y)=norm(x−y) is nonnegative, separates points, is symmetric and satisfies the triangle inequality. This slice uses exact coordinate calculations only for the named l1 and linfinity norms on R^n.





## Representative formula (exact TeX)

D(x,y)





## Assumptions


- Both points belong to the same finite-dimensional real coordinate space and use the same named norm.
- Induction of the metric follows from the universal formula and norm axioms, not a finite sample of point distances.





## Symbol scope


- d — norm-induced metric
- V — real normed vector space
- x — first vector
- y — second vector





## Intuition, definition, and misconceptions

Separation follows because norm(x−y)=0 iff x−y=0.

Symmetry follows from norm(−(y−x))=norm(y−x).

The triangle inequality follows from x−z=(x−y)+(y−z) and the norm triangle inequality.

Checking finitely many pairs of distances does not establish that an arbitrary metric is induced by some norm.





## Adopted derivation 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](https://ocme.evemisslab.com/en/derivations/derivation-mko-norm-induced-metric-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Real normed vector space](https://ocme.evemisslab.com/en/mko/mko-real-normed-vector-space/) — The induced formula relies on a certified norm.
- [Metric space](https://ocme.evemisslab.com/en/mko/mko-metric-space/) — The conclusion is the metric axioms.
- [Plane vectors in a fixed basis](https://ocme.evemisslab.com/en/mko/mko-vector/) — The formula uses a vector difference in the same space.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — The distance must be a total function from V×V to the nonnegative reals.





## Verification boundary

Only the induced formula for the same certified norm is handled; not every metric is claimed to be norm-induced, and completeness, an inner product, compactness or finite dimensionality is not inferred from the induced-metric property.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Together with complete metric spaces, support real Banach spaces.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/norm-induced-metric.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-norm-induced-metric/index.md)