# Real vector space

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

A real vector space has vector addition and real scalar multiplication, satisfying the additive abelian-group and compatible scalar-action axioms.

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



## Mathematical statement

A real vector space V has vector addition, a zero vector, additive inverses and real scalar multiplication, satisfying associativity, commutativity, distributivity, the unit-scalar law and scalar-multiplication compatibility. Exercises here check only exact axiom instances in fixed-dimensional real coordinate spaces.





## Representative formula (exact TeX)

V(R)





## Assumptions


- Exercises use equal fixed dimensions, integer coordinate vectors and integer scalars as special cases of real values.
- Successful coordinate instances do not fully prove an arbitrary candidate structure is a vector space.





## Symbol scope


- V — real vector space
- R — real scalar field





## Intuition, definition, and misconceptions

Vector addition is defined only between matching dimensions and adds corresponding coordinates.

Scalar multiplication scales each component. The zero vector and additive inverses support the additive-group structure.

Distributivity can be recomputed exactly for specified vectors and scalars; general axioms still quantify over all relevant elements.

Do not directly add vectors of different dimensions or present a few examples as a proof of general axioms.





## Adopted derivation guide

[Real vector space: auditing axiom instances in ℝ³ and proving one distributive law](https://ocme.evemisslab.com/en/derivations/derivation-mko-real-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


- [Plane vectors in a fixed basis](https://ocme.evemisslab.com/en/mko/mko-vector/) — Coordinate vectors supply a concrete model.
- [Real number](https://ocme.evemisslab.com/en/mko/mko-real-number/) — Scalars come from the real numbers.
- [Arithmetic operations](https://ocme.evemisslab.com/en/mko/mko-arithmetic-operations/) — Coordinate addition and scalar multiplication are componentwise.





## Verification boundary

Only a fixed R^n coordinate slice is checked; no arbitrary abstract structure is claimed to be a proved vector space.

not_formalized

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





## Next obligation

Add linear combinations, spans, independence, bases and dimension.





## 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/real-vector-space.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-real-vector-space/index.md)