Don't take these axioms too seriously!  Math is not about axioms, despite what some people say.  Axioms are one way to think precisely, but they are not the only way, and they are certainly not always the best way.  Also, there are a number of ways to phrase these axioms, and different books will do this differently, but they are all equivalent (unless the book author was really sloppy).

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 = x
• (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.

Axioms of a normed real vector space

A normed real vector space is a real vector space X with an additional operation:

• Norm: Given an element x in X, one can form the norm ||x||, which is a non-negative number.
This norm must satisfy the following axioms, for any x,y in X and any real number c:
• ||x|| = 0 if and only if x = 0.
• || cx || = |c| ||x||.
• || x+y || <= ||x|| + ||y||

Complex vector spaces and normed complex vector spaces are defined exactly as above, just replace every occurrence of "real" with "complex".  Note, though, that even in a complex vector space, the norm ||x|| is still a non-negative real number.
Thanks to Maxwell Davenport for a correction.