Question 13430: A Named-Norm Distance Is Not a Completeness Proof

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=[-8, 1, -4], right=[8, 1, -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=[-8, 1, -4], right=[8, 1, -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

Step-by-step derivation

Banach coordinate distance difference coordinate0 := sub(-8, 8)

Banach coordinate distance difference coordinate0 = -16

Banach coordinate distance difference coordinate1 := sub(1, 1)

Banach coordinate distance difference coordinate1 = 0

Banach coordinate distance difference coordinate2 := sub(-4, -2)

Banach coordinate distance difference coordinate2 = -2

Banach coordinate distance:norm=l1;left=[-8, 1, -4];right=[8, 1, -2]

signed left-right=[-16, 0, -2];use same named norm

Banach coordinate distance:coordinate=0;signed value=-16

absolute value=16

Banach coordinate distance:coordinate=1;signed value=0

absolute value=0

Banach coordinate distance:coordinate=2;signed value=-2

absolute value=2

Banach coordinate distance l1 sum := sum(16, 0, 2)

Banach coordinate distance l1 sum = 18

fixed carrier R^3;named norm=l1;distance=18

distance_evaluation_not_completeness_proof

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, [-8, 1, -4], [8, 1, -2], [-16, 0, -2], 18, 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.