Axioms of a vector space

Definition: A vector space over a field (\mathbb{F}) is a set (V) together with two operations --- vector addition ((+: V \times V \to V)) and scalar multiplication ((\cdot: \mathbb{F} \times V \to V)) --- satisfying the following ten axioms for all (\mathbf{u}, \mathbf{v}, \mathbf{w} \in V) and (a, b \in \mathbb{F}):

Addition axioms:

1. Closure: (\mathbf{u} + \mathbf{v} \in V)
2. Commutativity: (\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u})
3. Associativity: ((\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}))
4. Zero vector: There exists (\mathbf{0} \in V) such that (\mathbf{u} + \mathbf{0} = \mathbf{u})
5. Additive inverse: For each (\mathbf{u}), there exists (-\mathbf{u} \in V) such that (\mathbf{u} + (-\mathbf{u}) = \mathbf{0})

Scalar multiplication axioms:
6. Closure: (a \cdot \mathbf{u} \in V)
7. Distributivity over vectors: (a(\mathbf{u} + \mathbf{v}) = a\mathbf{u} + a\mathbf{v})
8. Distributivity over scalars: ((a + b)\mathbf{u} = a\mathbf{u} + b\mathbf{u})
9. Compatibility: (a(b\mathbf{u}) = (ab)\mathbf{u})
10. Identity: (1 \cdot \mathbf{u} = \mathbf{u})

Example: (\mathbb{R}^3) with standard operations satisfies all ten axioms and is therefore a vector space over (\mathbb{R}).