Axioms of real vector spaces(向量空間的8個公理)

Axioms of real vector spaces

A real vector space is a set X with a special element 0, and three operations:
Addition: Given two elements x, y in X, one can form the sum x+y, which is also an element of X.
Inverse: Given an element x in X, one can form the inverse -x, which is also an element of X.
Scalar multiplication: Given an element x in X and a real number c, one can form the product cx, which is also an element of X. These operations must satisfy the following axioms:
Additive axioms. For every x,y,z in X, we have
x+y = y+x.
(x+y)+z = x+(y+z).
0+x = x+0 = x.
(-x) + x = x + (-x) = 0.
Multiplicative axioms. For every x in X and real numbers c,d, we have
0x = 0
1x = 1
(cd)x = c(dx)
Distributive axioms. For every x,y in X and real numbers c,d, we have
c(x+y) = cx + cy.
(c+d)x = cx +dx.

http://www.math.ucla.edu/~tao/resource/general/121.1.00s/vector_axioms.html