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R0025-13453 · q-20261014-advance-20261028-r03-real-banach-space-0078
Use the explicit real coordinate space R^n of dimension=3, with the named norm=l1, 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. evidence_mode=finite_dimensional_theorem. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.
G001 — Use the explicit real coordinate space R^n of dimension=3, with the named norm=l1, 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. evidence_mode=finite_dimensional_theorem. Give (dimension,norm_name,evidence_mode,sufficient?,boundary). finite_dimensional_theorem yields universal_named_certificate; sampled_cauchy_sequences yields samples_insufficient_for_completeness.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.
Before
same real carrier R^3;standard coordinate vector operations;named norm=l1
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.
Rule: Banach_same_space_norm_metric_binding · Uses: G001
Before
arbitrary Cauchy sequence x_k in R^3 under l1;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.
Rule: Banach_coordinate_Cauchy_and_real_completeness_premise · Uses: G001, S001
Before
coordinate limits L_i form L in R^3;epsilon>0;each coordinate error eventually <epsilon/3
After
choose N=max of the 3 coordinate cutoffs;sum of 3 coordinate errors < 3*(epsilon/3)=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.
Rule: Banach_finite_coordinate_limits_same_norm_convergence · Uses: G001, S002
Before
every Cauchy sequence in this named real R^3 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.
Rule: named_finite_dimensional_Banach_certificate_scope · Uses: G001, S003
Before
named real space R^3;norm=l1;evidence=finite_dimensional_theorem
After
sufficient evidence=Yes;sample insufficiency is not incompleteness
Conditions: none
The named universal theorem and coordinate argument cover every Cauchy sequence;finite samples do not.No in sample mode neither declares this known space incomplete nor supplies a formal-verification receipt.
Rule: Banach_evidence_quantifier_scope · Uses: G001, S004
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.(3, l1, finite_dimensional_theorem, Yes, universal_named_certificate)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: e2072bfd6a7e2465584b09488d074d4d6727057d7fee28d93326cd52a051f334candidate record SHA-256: e1c961731df62f3d6b260b609b291aced2f2595ebb5ee9b23bbe064caa4a5932semantic review receipt SHA-256: 0e51c2bf621e6e998415fdd75d260464e625cecd5894a230c9f09553249551c5question-first witness SHA-256: ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8aLean: NOT_RUN · human review: NOT_PERFORMED