Real normed vector space

Mathematical statement

A norm sends vectors to nonnegative real numbers, with norm(v)=0 iff v=0, norm(a v)=|a| norm(v), and norm(u+v)≤norm(u)+norm(v). This slice uses only the named l1 and linfinity formulas on R^n.

Representative formula (exact TeX)

N(V)

Assumptions

• Exercises use integer-coordinate vectors of a fixed finite dimension.
• Qualification of l1 and linfinity comes from universal algebraic certificates for their formulas; finite samples do not stand in for general norm proofs.

Symbol scope

• N — norm record
• V — real vector space

Intuition, definition, and misconceptions

The l1 norm is the sum of the coordinate absolute values.

The linfinity norm is the largest coordinate absolute value.

Applying a norm to a vector difference induces a distance between two points; the induced metric is a separate node.

Passing nonnegativity and triangle tests at finitely many points does not establish that an arbitrary formula is a norm.

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 vector space — A norm is defined on a real vector space.
• Absolute value — Scalar homogeneity uses the absolute value of real scalars.
• Usual order on the real numbers — Nonnegativity and the triangle inequality use the real-number order.

Verification boundary

Only named exact formulas are accepted here; norm axioms are not established by sampling.

not_formalized

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

Next obligation

Add norm-induced metrics, Banach spaces and bounded operators.

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.