Inner-product-induced norm

Mathematical statement

In one real inner-product space define norm(x)=sqrt(<x,x>). Positive definiteness gives separation, homogeneity gives absolute homogeneity, and Cauchy–Schwarz gives the triangle inequality. This slice calculates exact squared norms and identities for positive-definite matrices on R² without replacing universal certificates with vector samples.

Representative formula (exact TeX)

N(J)

Assumptions

• The matrix already qualifies as bilinear, symmetric and positive definite, and the vectors lie in the same R².
• Square roots remain exact symbolic expressions; Cauchy–Schwarz is a universal certificate not established by finitely many vectors.

Symbol scope

• N — inner-product-induced norm
• J — same real inner product

Intuition, definition, and misconceptions

Compute the squared norm <x,x> with exact integer arithmetic first.

The squared distance is <x−y,x−y>.

Cauchy–Schwarz and the parallelogram identity check the structure but do not imply completeness.

A few successful triangle-inequality samples do not certify a general norm, and an inner-product-induced norm alone does not make a space Hilbert.

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 — The formula uses the same certified inner product.
• Natural-number powers and principal square roots — The norm is the nonnegative square root of the self-inner-product.
• Usual order on the real numbers — Positive definiteness and Cauchy–Schwarz use the real ordering.

Verification boundary

Only the explicit R² inner-product matrix and exact certificates are checked; completeness and Hilbert structure 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

Combine with completeness to add 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.