Norm-induced metric

Mathematical statement

If (V,norm) is a certified real normed vector space, d(x,y)=norm(x−y) is nonnegative, separates points, is symmetric and satisfies the triangle inequality. This slice uses exact coordinate calculations only for the named l1 and linfinity norms on R^n.

Representative formula (exact TeX)

D(x,y)

Assumptions

• Both points belong to the same finite-dimensional real coordinate space and use the same named norm.
• Induction of the metric follows from the universal formula and norm axioms, not a finite sample of point distances.

Symbol scope

• d — norm-induced metric
• V — real normed vector space
• x — first vector
• y — second vector

Intuition, definition, and misconceptions

Separation follows because norm(x−y)=0 iff x−y=0.

Symmetry follows from norm(−(y−x))=norm(y−x).

The triangle inequality follows from x−z=(x−y)+(y−z) and the norm triangle inequality.

Checking finitely many pairs of distances does not establish that an arbitrary metric is induced by some 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 normed vector space — The induced formula relies on a certified norm.
• Metric space — The conclusion is the metric axioms.
• Plane vectors in a fixed basis — The formula uses a vector difference in the same space.
• Function mapping — The distance must be a total function from V×V to the nonnegative reals.

Verification boundary

Only the induced formula for the same certified norm is handled; not every metric is claimed to be norm-induced, and completeness, an inner product, compactness or finite dimensionality is not inferred from the induced-metric property.

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 complete metric spaces, support real Banach 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.