Real Hilbert space

Mathematical statement

The carrier, inner product, induced norm, induced metric and completeness certificate must describe the same space. This slice uses named completeness-theorem certificates for finite-dimensional R^n with a standard or positive-definite-matrix inner product. Sampled Cauchy sequences do not establish completeness.

Representative formula (exact TeX)

H(V)

Assumptions

• The finite-dimensional R^n and its matching positive-definite inner-product matrix are explicit.
• Completeness uses the theorem for finite-dimensional real inner-product spaces, not sampled sequences.

Symbol scope

• H — real Hilbert space
• V — same real inner-product space

Intuition, definition, and misconceptions

All five structures must refer to the same source space.

Finite-dimensional R^n provides a named completeness certificate.

Hilbert structure alone is not used to assert compactness, separability or arbitrary-dimensional conclusions.

Mixing unrelated norms and metrics, or checking finitely many Cauchy sequences, does not establish the required completeness binding.

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 inner-product space — A Hilbert space must first be an inner-product space.
• Inner-product-induced norm — The norm comes from that same inner product.
• Norm-induced metric — The metric comes from that same norm.
• Complete metric space — Every Cauchy sequence converges within that same space.

Verification boundary

Only explicitly specified finite-dimensional certificates are covered; no compactness, separability or infinite-dimensional result is inferred. The active question batch now uses the owner-adopted matrix-aligned two-dimensional revision of 2026-10-17; the original mismatched batch remains preserved as history.

not_formalized

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

Next obligation

Add orthogonal projections and Hilbert bases.

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.