Use the explicit real coordinate space R^n of dimension=21, 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. metric_source=unrelated_metric, completeness_source=sampled_sequences. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.
G001 — Use the explicit real coordinate space R^n of dimension=21, 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. metric_source=unrelated_metric, completeness_source=sampled_sequences. Give (dimension,norm_name,metric_source,completeness_source,metric_same?,complete?,Banach?). Recognized certificate labels are same_norm_induced and finite_dimensional_theorem.
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.
Each flag records an exact source-label match to same_norm_induced or finite_dimensional_theorem,then conjunction.A missing binding makes this record unqualified;it does not refute the finite-dimensional completeness theorem.
S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
Result
(21, l1, unrelated_metric, sampled_sequences, No, No, No)
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.