# Real Banach space

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

A real normed vector space is a Banach space when it is complete in the metric induced by its own norm.

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



## Mathematical statement

A real Banach space binds one carrier, its real vector operations, its norm, the metric induced by that norm and completeness into one structure. This slice uses exact named-theorem certificates for finite-dimensional R^n with l1 or linfinity, not sampled sequences as completeness proofs.





## Representative formula (exact TeX)

B(V)





## Assumptions


- The space is an explicit finite-dimensional R^n with the named l1 or linfinity norm; the metric must come from that same norm.
- The completeness certificate invokes the named theorem for finite-dimensional real normed spaces; finitely many Cauchy-sequence samples are insufficient.





## Symbol scope


- B — real Banach space
- V — same real normed space





## Intuition, definition, and misconceptions

The norm and metric must not come from different spaces or different formulas.

Named norms on finite-dimensional R^n supply a completeness-theorem certificate.

A Banach space does not automatically have an inner product or Hilbert structure.

Testing a few Cauchy sequences, or knowing that some metric is complete, does not establish an arbitrary normed space as Banach.





## Adopted derivation guide

[Real Banach space: norm, induced metric and completeness of (ℝ², ℓ¹)](https://ocme.evemisslab.com/en/derivations/derivation-mko-real-banach-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 normed vector space](https://ocme.evemisslab.com/en/mko/mko-real-normed-vector-space/) — A Banach space is first a real normed space.
- [Norm-induced metric](https://ocme.evemisslab.com/en/mko/mko-norm-induced-metric/) — Completeness uses the metric induced by the same norm.
- [Complete metric space](https://ocme.evemisslab.com/en/mko/mko-complete-metric-space/) — Every Cauchy sequence must converge inside the space.
- [Real vector space](https://ocme.evemisslab.com/en/mko/mko-real-vector-space/) — The carrier and operations must describe the same real vector space.





## Verification boundary

The named certificate covers only the explicit finite-dimensional R^n and the same norm; it does not infer Hilbert structure, compactness or arbitrary infinite-dimensional conclusions.

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