Dual vector space

Mathematical statement

This slice uses finite-dimensional V=R^n. A row covector gives a symbolic linearity certificate, and a dual basis is checked by phi_i(v_j)=delta_ij. A primal vector and a dual functional are distinct; without extra structure no canonical identification V→V* is claimed.

Representative formula (exact TeX)

\varphi\in V^*

Assumptions

• Fix a finite-dimensional real space and an explicit ordered basis.
• Use exact rational arithmetic for matrices, functional values and changes of basis.

Symbol scope

• V^* — dual vector space
• \varphi — linear functional
• V — finite real vector space

Intuition, definition, and misconceptions

A dual element is a function, not a primal vector.

Every dual-basis pairing must satisfy its Kronecker-delta condition.

Primal and covector coordinates follow opposite change laws; identifying their spaces requires a choice of extra structure.

Notation candidate: V* is the real dual space of linear functionals V→R; the representative formula declares membership,not an unspecified power or a canonical identification of vectors and functionals.

Equal-length coordinate arrays do not make vectors and functionals the same canonical object.

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

• Linear map between real vector spaces — A linear functional is a linear map to R.
• Real vector space — The source space and scalar field are fixed.
• Real number — Functional values lie in R.

Verification boundary

Only finite-dimensional R^n and explicit bases are handled; no infinite-dimensional continuous-dual or canonical-isomorphism claim is 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 annihilators, the double dual and adjoints.

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.