DEFINITION OF VECTOR SPACE

(0) V  is a non-empty set and  K  is a field.
(i)  (internal binary operation, "addition")
Å is a function V×V ® V, i.e. for each (u,v) Î V×V there is exactly one element
  u Å v : =  Å(u,v) Î V.
(ii) (scaling function, "multiplication by scalars")
· is a function  K×V ® V, i.e. for each (a,u) Î K×V there is exactly one element
    a·u : = ·(a,u) Î V.
(iii) For addition and multiplication by scalars the following Axioms hold:
A1. u Å v = v Å u                                          for all  u, v Î V (commutativity).
A2. (u Å v) Å w = u Å (v Å w)                     for all u, v, w Î V (associativity).
A3. There exists an element e Î V, for which u Å e = u and  e Å u = u for all u Î V
(existence of neutral or null element e).
A4. For every u Î V there is  -u Î V, for which u Å (-u) = e  and  (-u) Åu = e
(existence of inverse or negative of u).
A5. a·(u Å v) = a·u Å a·v                            for all a Î K, for all  u, v Î V.
A6. (a+b)·u = a·u Å b·u                              for all a, b Î K, for all  u Î V.
A7. a·(b·u) = (ab) ·u                                    for all a, b Î K, for all u Î V.
A8. 1·u = u                                                    for all  u Î V.

If (V,Å,·) is a K-vector space, the field K is called the scalar space or scalar field.
If K = R, then (V,Å,·) is a real vector space; if K = C, then (V,Å,·) is a complex vector space.
In what follows we talk simply about vector space V instead of a triple.