# Dual vector space

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

The dual of a real vector space consists of all real-valued linear functionals; a covector coordinate row depends on a chosen basis.

mko-dual-vector-space · definition · v0.10.0



## 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

[Dual basis and covector change](https://ocme.evemisslab.com/en/derivations/derivation-mko-dual-vector-space-01/)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](https://ocme.evemisslab.com/en/mko/mko-general-linear-map/) — A linear functional is a linear map to R.
- [Real vector space](https://ocme.evemisslab.com/en/mko/mko-real-vector-space/) — The source space and scalar field are fixed.
- [Real number](https://ocme.evemisslab.com/en/mko/mko-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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/source13-20261018-dual-vector-space.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-dual-vector-space/index.md)