# Question 13489: A Named Finite-Dimensional Banach Certificate Binds One Space

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0025-13489 · q-20261014-advance-20261028-r03-real-banach-space-0014



## Original question and givens

Use the explicit real coordinate space R^n of dimension=14, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).


- G001 — Use the explicit real coordinate space R^n of dimension=14, with the named norm=linfinity, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. Give (dimension,norm_name,R^n,same_norm_induced_metric,finite_dimensional_real_normed_space_complete,Banach?,not_Hilbert_claim).





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-real-banach-space](https://ocme.evemisslab.com/en/mko/mko-real-banach-space/)





## Step-by-step derivation

S001 · impliesBefore

same real carrier R^14;standard coordinate vector operations;named norm=linfinity

After

metric is d(x,y)=this_same_norm(x-y);norm axioms supplied by named formula

Conditions: none

Carrier,vector operations,norm,induced metric and completeness must describe the same space.A completeness result for another metric cannot be transplanted.

Step identifiers and dependenciesRule: Banach_same_space_norm_metric_binding · Uses: G001





S002 · impliesBefore

arbitrary Cauchy sequence x_k in R^14 under linfinity;every coordinate obeys abs((x_k-x_m)_i)<=norm(x_k-x_m)

After

each real coordinate sequence is Cauchy;real-number completeness supplies a real limit L_i

Conditions: none

For every positive eta,the vector Cauchy tail bound controls every coordinate difference,so each coordinate is Cauchy.Real-number completeness is an explicit mathematical premise here,not a new Lean verification.

Step identifiers and dependenciesRule: Banach_coordinate_Cauchy_and_real_completeness_premise · Uses: G001, S001





S003 · impliesBefore

coordinate limits L_i form L in R^14;epsilon>0;each coordinate error eventually <epsilon

After

choose N=max of the 14 coordinate cutoffs;maximum of finitely many coordinate errors < epsilon;norm(x_k-L)->0

Conditions: none

Finite positive dimension permits one maximum cutoff for all coordinates.Use epsilon/n for l1 and the finite maximum for linfinity.This gives convergence in the same norm to L in the original space,not merely coordinatewise or ambient convergence.

Step identifiers and dependenciesRule: Banach_finite_coordinate_limits_same_norm_convergence · Uses: G001, S002





S004 · impliesBefore

every Cauchy sequence in this named real R^14 normed space has a same-space norm limit

After

Banach certificate=Yes;no Hilbert or arbitrary infinite-dimensional conclusion

Conditions: none

The universal argument closes completeness of this named finite-dimensional space.It does not infer an inner product,Hilbert structure,compactness or arbitrary infinite-dimensional conclusions.

Step identifiers and dependenciesRule: named_finite_dimensional_Banach_certificate_scope · Uses: G001, S003









## Reverse checks


- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(14, linfinity, R^n, same_norm_induced_metric, finite_dimensional_real_normed_space_complete, Yes, not_Hilbert_claim)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 0b6539d22d54a70d00c7fafc7878fa912a56c21a519875437b032352026586bb
- candidate record SHA-256: f01c4c3fe2e25fe8177f0630c1362ac5ae7040600e206be88eb73470c4432daf
- semantic review receipt SHA-256: 0e51c2bf621e6e998415fdd75d260464e625cecd5894a230c9f09553249551c5
- question-first witness SHA-256: ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8a
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261028-r03-real-banach-space-0014.json)