# Inner-product-induced norm

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

On the same real inner-product space, the norm is the nonnegative square root of the self-inner-product; the triangle inequality uses the universal Cauchy–Schwarz certificate.

mko-inner-product-induced-norm · definition · v0.10.0



## Mathematical statement

In one real inner-product space define norm(x)=sqrt(<x,x>). Positive definiteness gives separation, homogeneity gives absolute homogeneity, and Cauchy–Schwarz gives the triangle inequality. This slice calculates exact squared norms and identities for positive-definite matrices on R² without replacing universal certificates with vector samples.





## Representative formula (exact TeX)

N(J)





## Assumptions


- The matrix already qualifies as bilinear, symmetric and positive definite, and the vectors lie in the same R².
- Square roots remain exact symbolic expressions; Cauchy–Schwarz is a universal certificate not established by finitely many vectors.





## Symbol scope


- N — inner-product-induced norm
- J — same real inner product





## Intuition, definition, and misconceptions

Compute the squared norm <x,x> with exact integer arithmetic first.

The squared distance is <x−y,x−y>.

Cauchy–Schwarz and the parallelogram identity check the structure but do not imply completeness.

A few successful triangle-inequality samples do not certify a general norm, and an inner-product-induced norm alone does not make a space Hilbert.





## Adopted derivation guide

[Inner-product-induced norm: proving √⟨x,x⟩ is a norm and computing exactly for a positive-definite matrix on ℝ²](https://ocme.evemisslab.com/en/derivations/derivation-mko-inner-product-induced-norm-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Real inner-product space](https://ocme.evemisslab.com/en/mko/mko-real-inner-product-space/) — The formula uses the same certified inner product.
- [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/mko/mko-power-root/) — The norm is the nonnegative square root of the self-inner-product.
- [Usual order on the real numbers](https://ocme.evemisslab.com/en/mko/mko-order-relation/) — Positive definiteness and Cauchy–Schwarz use the real ordering.





## Verification boundary

Only the explicit R² inner-product matrix and exact certificates are checked; completeness and Hilbert structure are not inferred.

not_formalized

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





## Next obligation

Combine with completeness to add real Hilbert 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/inner-product-induced-norm.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-inner-product-induced-norm/index.md)