In this chapter we introduce the objects we will be studying and investigate some of their basic properties.

BASIC DEFINITIONS AND EXAMPLES

Definition 1.1.1. A vector space V over a field F is a set V with a pair of operations (u, v) ↦ u + v for u, v ∈ V and (c, u) ↦ cu for c ∈ F, v ∈ V satisfying the following axioms:

1. (1) u + v ∈ V for any u, v ∈ V.

2. (2) u + v = v + u for any u, v ∈ V.

3. (3) u + (v + w) = (u + v) + w for any u, v, w ∈ V.

4. (4) There is a 0 ∈ V such that 0 + v = v + 0 = v for any v ∈ V.

5. (5) For any v ∈ V there is a -v ∈ V such that v + (-v) = (-v) + v = 0.

6. (6) cv ∈ V for any c ∈ F, v ∈ V.

7. (7) c(u + v) = cu + cv for any c ∈ F, u, v ∈ V.

8. (8) (c + d)u = cu + du for any c, d ∈ F, u ∈ V.

9. (9) c(du) = (cd)u for any c, d ∈ F, u ∈ V.

10. (10) 1u = u for any u ∈ V.