# Real normed vector space

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

A real vector space equipped with a positive-definite, absolutely homogeneous real-valued norm satisfying the triangle inequality.

mko-real-normed-vector-space · definition · v0.10.0



## Mathematical statement

A norm sends vectors to nonnegative real numbers, with norm(v)=0 iff v=0, norm(a v)=|a| norm(v), and norm(u+v)≤norm(u)+norm(v). This slice uses only the named l1 and linfinity formulas on R^n.





## Representative formula (exact TeX)

N(V)





## Assumptions


- Exercises use integer-coordinate vectors of a fixed finite dimension.
- Qualification of l1 and linfinity comes from universal algebraic certificates for their formulas; finite samples do not stand in for general norm proofs.





## Symbol scope


- N — norm record
- V — real vector space





## Intuition, definition, and misconceptions

The l1 norm is the sum of the coordinate absolute values.

The linfinity norm is the largest coordinate absolute value.

Applying a norm to a vector difference induces a distance between two points; the induced metric is a separate node.

Passing nonnegativity and triangle tests at finitely many points does not establish that an arbitrary formula is a norm.





## Adopted derivation guide

[Real normed vector space: universal norm certificates for the ℓ¹ and ℓ^∞ formulas, and two formulas that are not norms](https://ocme.evemisslab.com/en/derivations/derivation-mko-real-normed-vector-space-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Real vector space](https://ocme.evemisslab.com/en/mko/mko-real-vector-space/) — A norm is defined on a real vector space.
- [Absolute value](https://ocme.evemisslab.com/en/mko/mko-absolute-value/) — Scalar homogeneity uses the absolute value of real scalars.
- [Usual order on the real numbers](https://ocme.evemisslab.com/en/mko/mko-order-relation/) — Nonnegativity and the triangle inequality use the real-number order.





## Verification boundary

Only named exact formulas are accepted here; norm axioms are not established by sampling.

not_formalized

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





## Next obligation

Add norm-induced metrics, Banach spaces and bounded operators.





## 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/real-normed-vector-space.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-real-normed-vector-space/index.md)