Real Banach space

Mathematical statement

A real Banach space binds one carrier, its real vector operations, its norm, the metric induced by that norm and completeness into one structure. This slice uses exact named-theorem certificates for finite-dimensional R^n with l1 or linfinity, not sampled sequences as completeness proofs.

Representative formula (exact TeX)

B(V)

Assumptions

• The space is an explicit finite-dimensional R^n with the named l1 or linfinity norm; the metric must come from that same norm.
• The completeness certificate invokes the named theorem for finite-dimensional real normed spaces; finitely many Cauchy-sequence samples are insufficient.

Symbol scope

• B — real Banach space
• V — same real normed space

Intuition, definition, and misconceptions

The norm and metric must not come from different spaces or different formulas.

Named norms on finite-dimensional R^n supply a completeness-theorem certificate.

A Banach space does not automatically have an inner product or Hilbert structure.

Testing a few Cauchy sequences, or knowing that some metric is complete, does not establish an arbitrary normed space as Banach.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Real normed vector space — A Banach space is first a real normed space.
• Norm-induced metric — Completeness uses the metric induced by the same norm.
• Complete metric space — Every Cauchy sequence must converge inside the space.
• Real vector space — The carrier and operations must describe the same real vector space.

Verification boundary

The named certificate covers only the explicit finite-dimensional R^n and the same norm; it does not infer Hilbert structure, compactness or arbitrary infinite-dimensional conclusions.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Together with inner-product-induced norms, support real Hilbert spaces.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.