Abstract Vector Space

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.

REFERENCES:

Peano, G. Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann. Torino, Italia: Fratelli Bocca, 1888.