LINEAR SPACE (ABSTRACT VECTOR SPACE)

The definition of an abstract vector space is an obscure, abstruse, axiomatic one as are the definitions of many concepts of modern, abstract, axiom-based mathematics. A set of abstract algebraic laws or properties are stated as postulates that must be satisfied in order for the system to qualify as a vector space. These postulates derive from a list of properties of n-dimensional space.

Def. Linear Space (or Abstract Vector Space). A linear space is a set V of elements, called vectors, of an arbitrary nature for which the concepts of a sum and scalar multiplication (i.e. a product by a number) make sense, which is closed with respect to addition and scalar multiplication, and which satisfy the following postulates for any element x,y and z in V and any scalars a,b and c in a number field F:

  1. (x+y) + z = x + (y+z)

  2. There exists a "null" element 0 such that x+0 = x for every x in V

  3. For every element x there exists an opposite -x such that x+(-x) = 0

  4. x + y = y + x

  5. 1x = x for the unit 1 in F and for all x

  6. a(bx) = (ab)x

  7. (a+b)x = ax + bx

  8. c(x+y) = cx + cy

Postulates 1 through 4 are equivalent to the statement that V is an Abelian group with addition as the group operation.