Examples of Vector Spaces - Infinite Coordinate Space

Infinite Coordinate Space

Let F∞ denote the space of infinite sequences of elements from F such that only finitely many elements are nonzero. That is, if we write an element of F∞ as

then only a finite number of the x_i are nonzero (i.e., the coordinates become all zero after a certain point). Addition and scalar multiplication are given as in finite coordinate space. The dimensionality of F∞ is countably infinite. A standard basis consists of the vectors e_i which contain a 1 in the i-th slot and zeros elsewhere. This vector space is the coproduct (or direct sum) of countably many copies of the vector space F.

Note the role of the finiteness condition here. One could consider arbitrary sequences of elements in F, which also constitute a vector space with the same operations, often denoted by FN - see below. FN is the product of countably many copies of F.

By Zorn's lemma, FN has a basis (there is no obvious basis). There are uncountably infinite elements in the basis. Since the dimensions are different, FN is not isomorphic to F∞. It is worth noting that FN is (isomorphic to) the dual space of F∞, because a linear map T from F∞ to F is determined uniquely by its values T(e_i) on the basis elements of F∞, and these values can be arbitrary. Thus one sees that a vector space need not be isomorphic to its dual if it is infinite dimensional, in contrast to the finite dimensional case.