There are at least two distinct notions of linear space throughout mathematics.
The term linear space is most commonly used within functional analysis as a synonym of the term vector space.
The term is also used to describe a fundamental notion in the field of incidence geometry. In particular, a linear space is a space 
consisting of a collection 
of points and a set 
of lines
subject to the following axioms:
1. Any two distinct points of 
belong to exactly one line of 
.
2. Any line of 
has at least two points of 
.
3. There are at least three points of 
not on a common line.
Using this terminology, lines are considered to be "distinguished subsets" of the collection 
of points. Moreover, in this context, one can view a linear space as a generalization
of the notions of projective space and affine
space (Batten and Beutelspracher 2009).
