An abstract vector space of dimension 
over a field 
is the set of all formal expressions
 |
(1)
|
where 
is a given set of 
objects (called a basis) and 
is any 
-tuple of elements of 
. Two such expressions can be added together by summing their
coefficients,
 |
(2)
|
This addition is a commutative group operation, since the zero element is 
and the inverse of 
is 
. Moreover, there is a natural
way to define the product of any element 
by an arbitrary element (a so-called
scalar) 
of 
,
 |
(3)
|
Note that multiplication by 1 leaves the element unchanged.
This structure is a formal generalization of the usual vector space over 
,
for which the field of scalars is the real field 
and a basis is given by 
. As in this
special case, in any abstract vector space 
, the multiplication by scalars fulfils the following two distributive
laws:
1. For all 
and all 
,
.
2. For all 
and all 
,
.
These are the basic properties of the integer multiples in any commutative additive group. This special behavior of a product with respect to the sum defines the notion of linear structure, which was first formulated by Peano in 1888.
Linearity implies, in particular, that the zero elements 
and 
of 
and 
annihilate any product. From (1), it follows that
 |
(4)
|
for all 
,
whereas from (2), it follows that
 |
(5)
|
for all 
.
A more general kind of abstract vector space is obtained if one admits that the basis has infinitely many elements. In this case, the vector space is called infinite-dimensional and its elements are the formal expressions in which all but a finite number of coefficients are equal to zero.
