A vector space over a field F is a set V (whose elements are called vectors) together with a field F (whose elements are called scalars) , equipped with two operations:

1. Vector Addition: An operation that takes two vectors `bb {u} , bb{v} in V` and returns a vector `bb {u + v} in V`.
2. Scalar Multiplication: An operation that takes a scalar `a in F` and a vector `bb {u } in V` and returns a vector `a bb{u} in V`.

Note: The set V and the field F are distinct entities.
The set V itself must satisfy the following ten axioms:

II. Axioms for Scalar Multiplication (Interaction between F and W)
The operation of scalar multiplication must satisfy the following five conditions.
For all scalars `a , b in F` and all vectors `bb {u , v} in V`:

6. Closure under Scalar Multiplication: `a bb{u} in V`.
7. Associativity of Scalar Multiplication: `a(b bb{u}) = (ab) bb{u}`.
8. Distributivity over Vector Addition: `a(bb {u + v}) = a bb{u} + a bb{v}`.
9. Distributivity over Scalar Addition: `(a + b) bb{u} = a bb{u} + b bb{u}`.
10. Multiplicative identity 1: `1 bb{u} = bb{u}`.

Clarification on V and F:
- V is the abelian group: V is the set of vectors , and is endowed with the abelian group structure via vector addition (Axiom 1-5).
- F is the field: F is a seperate algebraic object that provides the scalars.
- The vector space itself is the entire structure - the ordered triple `bb{(V , F , *) }` , where `*` denotes the scalar multiplication operation. We often simply refer to the vector space as V for brevity , but mathematically , it's defined by the interaction between the set V and the field F. Axiom 6 `(a bb{u} in V)` simply confirms that scalar multiplication is a closed operation that keeps the results within the set of vectors V.