# Real inner-product space

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

A symmetric, positive-definite real bilinear form is a real inner product; positive semidefiniteness or positive sampled values do not suffice.

mko-real-inner-product-space · definition · v0.10.0



## Mathematical statement

On the same real vector space, an inner product must be bilinear in both slots, symmetric and satisfy <x,x>>0 for every nonzero x. This slice uses integer-coefficient symmetric matrices on R² and the Sylvester conditions as a universal positive-definiteness certificate.





## Representative formula (exact TeX)

J(V)





## Assumptions


- The matrix and vectors are fully specified, with a candidate 2-by-2 integer matrix.
- Positive definiteness is checked by exact leading-principal-minor and determinant conditions, not established by finitely many sampled vectors.





## Symbol scope


- J — real inner product
- V — real vector space





## Intuition, definition, and misconceptions

Bilinearity, symmetry and positive definiteness are three separate obligations.

A symmetric matrix on R² admits a universal Sylvester certificate.

An inner-product space is not automatically complete.

A symmetric bilinear or positive-semidefinite form is not automatically an inner product; a few positive vector samples do not prove positive definiteness.





## Adopted derivation guide

[Real inner-product space: Sylvester conditions as a universal positive-definiteness certificate for a matrix form on ℝ²](https://ocme.evemisslab.com/en/derivations/derivation-mko-real-inner-product-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 bilinear form](https://ocme.evemisslab.com/en/mko/mko-real-bilinear-form/) — An inner product must first be fully bilinear.
- [Usual order on the real numbers](https://ocme.evemisslab.com/en/mko/mko-order-relation/) — Positive definiteness uses strict positivity of real values.
- [Real vector space](https://ocme.evemisslab.com/en/mko/mko-real-vector-space/) — The inner product is defined on the same real vector space.





## Verification boundary

Only the explicit R² matrix certificate is checked; completeness, Hilbert structure and a spectral theorem in arbitrary dimension 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

Add inner-product-induced norms.





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