# Question 13434: A Named-Norm Distance Is Not a Completeness Proof [Return to catalogue page 68](https://ocme.evemisslab.com/en/questions/page/68/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261028-r03-real-banach-space-foundations/#q-20261014-advance-20261028-r03-real-banach-space-0059) 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. R0025-13434 · q-20261014-advance-20261028-r03-real-banach-space-0059 ## Original question and givens 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. left=[-4, -2, -3], right=[4, -5, -2]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof). - 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. left=[-4, -2, -3], right=[4, -5, -2]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof). ## 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 · calculationBefore Banach coordinate distance difference coordinate0 := sub(-4, 4) After Banach coordinate distance difference coordinate0 = -8 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · calculationBefore Banach coordinate distance difference coordinate1 := sub(-2, -5) After Banach coordinate distance difference coordinate1 = 3 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Banach coordinate distance difference coordinate2 := sub(-3, -2) After Banach coordinate distance difference coordinate2 = -1 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore Banach coordinate distance:norm=l1;left=[-4, -2, -3];right=[4, -5, -2] After signed left-right=[-8, 3, -1];use same named norm Conditions: none Use the same stated norm on the same real coordinate space.The derived difference can exceed original input coordinate bounds and must not be clipped or rejected as a new raw input. Step identifiers and dependenciesRule: norm_distance_same_space_and_direction · Uses: G001, S003 S005 · impliesBefore Banach coordinate distance:coordinate=0;signed value=-8 After absolute value=8 Conditions: none Take each whole coordinate's absolute value while preserving the original signed vector.Do not use a signed maximum. Step identifiers and dependenciesRule: named_norm_coordinate_absolute_value · Uses: G001, S004 S006 · impliesBefore Banach coordinate distance:coordinate=1;signed value=3 After absolute value=3 Conditions: none Take each whole coordinate's absolute value while preserving the original signed vector.Do not use a signed maximum. Step identifiers and dependenciesRule: named_norm_coordinate_absolute_value · Uses: G001, S005 S007 · impliesBefore Banach coordinate distance:coordinate=2;signed value=-1 After absolute value=1 Conditions: none Take each whole coordinate's absolute value while preserving the original signed vector.Do not use a signed maximum. Step identifiers and dependenciesRule: named_norm_coordinate_absolute_value · Uses: G001, S006 S008 · calculationBefore Banach coordinate distance l1 sum := sum(8, 3, 1) After Banach coordinate distance l1 sum = 12 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore fixed carrier R^3;named norm=l1;distance=12 After distance_evaluation_not_completeness_proof Conditions: none One distance evaluation quantifies over no Cauchy sequences and is not itself a completeness proof. Step identifiers and dependenciesRule: Banach_distance_evaluation_not_all_Cauchy_claim · Uses: G001, S008 ## Reverse checks - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Recompute Banach coordinate distance l1 sum with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Recompute Banach coordinate distance difference coordinate2 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Banach coordinate distance difference coordinate1 with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Recompute Banach coordinate distance difference coordinate0 with exact rational arithmetic, not rounded decimals. ## Result (l1, [-4, -2, -3], [4, -5, -2], [-8, 3, -1], 12, distance_evaluation_not_completeness_proof) 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: f599bf981e7b675830b40411e7d9ea0410606410ff646acc0f8acb7d7e4a0d45 - candidate record SHA-256: 0b5fd0f253a21b0c4cad5728643847770f6bfab43414337731cfc6c0cb7b8269 - 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-0059.json)