Real inner-product space

Mathematical statement

On the same real vector space, an inner product must be bilinear in both slots, symmetric and satisfy <x,x>>0 for every nonzero x. This slice uses integer-coefficient symmetric matrices on R² and the Sylvester conditions as a universal positive-definiteness certificate.

Representative formula (exact TeX)

J(V)

Assumptions

• The matrix and vectors are fully specified, with a candidate 2-by-2 integer matrix.
• Positive definiteness is checked by exact leading-principal-minor and determinant conditions, not established by finitely many sampled vectors.

Symbol scope

• J — real inner product
• V — real vector space

Intuition, definition, and misconceptions

Bilinearity, symmetry and positive definiteness are three separate obligations.

A symmetric matrix on R² admits a universal Sylvester certificate.

An inner-product space is not automatically complete.

A symmetric bilinear or positive-semidefinite form is not automatically an inner product; a few positive vector samples do not prove positive definiteness.

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 bilinear form — An inner product must first be fully bilinear.
• Usual order on the real numbers — Positive definiteness uses strict positivity of real values.
• Real vector space — The inner product is defined on the same real vector space.

Verification boundary

Only the explicit R² matrix certificate is checked; completeness, Hilbert structure and a spectral theorem in arbitrary dimension are not inferred.

not_formalized

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

Next obligation

Add inner-product-induced norms.

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.