# Real Hilbert space

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

A real inner-product space is Hilbert when it is complete in the norm and metric induced by that same inner product.

mko-real-hilbert-space · definition · v0.10.0



## Mathematical statement

The carrier, inner product, induced norm, induced metric and completeness certificate must describe the same space. This slice uses named completeness-theorem certificates for finite-dimensional R^n with a standard or positive-definite-matrix inner product. Sampled Cauchy sequences do not establish completeness.





## Representative formula (exact TeX)

H(V)





## Assumptions


- The finite-dimensional R^n and its matching positive-definite inner-product matrix are explicit.
- Completeness uses the theorem for finite-dimensional real inner-product spaces, not sampled sequences.





## Symbol scope


- H — real Hilbert space
- V — same real inner-product space





## Intuition, definition, and misconceptions

All five structures must refer to the same source space.

Finite-dimensional R^n provides a named completeness certificate.

Hilbert structure alone is not used to assert compactness, separability or arbitrary-dimensional conclusions.

Mixing unrelated norms and metrics, or checking finitely many Cauchy sequences, does not establish the required completeness binding.





## Adopted derivation guide

[A finite-dimensional Hilbert certificate from one matrix](https://ocme.evemisslab.com/en/derivations/derivation-mko-real-hilbert-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 inner-product space](https://ocme.evemisslab.com/en/mko/mko-real-inner-product-space/) — A Hilbert space must first be an inner-product space.
- [Inner-product-induced norm](https://ocme.evemisslab.com/en/mko/mko-inner-product-induced-norm/) — The norm comes from that same inner product.
- [Norm-induced metric](https://ocme.evemisslab.com/en/mko/mko-norm-induced-metric/) — The metric comes from that same norm.
- [Complete metric space](https://ocme.evemisslab.com/en/mko/mko-complete-metric-space/) — Every Cauchy sequence converges within that same space.





## Verification boundary

Only explicitly specified finite-dimensional certificates are covered; no compactness, separability or infinite-dimensional result is inferred. The active question batch now uses the owner-adopted matrix-aligned two-dimensional revision of 2026-10-17; the original mismatched batch remains preserved as history.

not_formalized

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





## Next obligation

Add orthogonal projections and Hilbert bases.





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