Some basic definitions

Consider a collection C of elements or “vectors” with the following properties:

1. They can be added. If f and g are in C, so is f + g.

2. f + (g + h) = (f + g) + h, f, g, h ∈ C.

3. There is an element 0 ∈ C such that h + 0 = h for all h ∈ C.

4. For each h ∈ C there is an element −h ∈ C such that h + (− h) = 0.

5. g + h = h + g, g, h ∈ C.

6. For each real number α, αh ∈ C.

7. α(g + h) = αg + αh.

8. (α + β)h = α h + β h.

9. α (βh) = (α β)h.

10. To each h ∈ C there corresponds a real number ∥h∥ with the following properties:

11. ∥αh∥ = |α| ∥h∥.

12. ∥h∥ = 0 if, and only if, h = 0.

13. ∥g + h∥ ≤ ∥g∥ + ∥h∥.

14. If {hn} is a sequence of elements of C such that ∥hn − hm∥ → 0 as m, n → ∞, then there is an element h ∈ C such that ∥hn − h∥ → 0 as n → ∞.