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.

- 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.

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.

- ||x|| = 0 if and only if x = 0.
- || cx || = |c| ||x||.
- || x+y || <= ||x|| + ||y||

Thanks to Maxwell Davenport for a correction.