Editing Examples of vector spaces (section)

==Infinite coordinate space==
Let ''F''<sup>∞</sup> denote the space of [[infinite sequence]]s of elements from ''F'' such that only ''finitely'' many elements are nonzero. That is, if we write an element of ''F''<sup>∞</sup> as
:<math>x = (x_1, x_2, x_3, \ldots) </math>
then only a finite number of the ''x''<sub>''i''</sub> 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''<sup>∞</sup> is [[countably infinite]]. A standard basis consists of the vectors ''e''<sub>''i''</sub> which contain a 1 in the ''i''-th slot and zeros elsewhere.  This vector space is the [[coproduct]] (or [[direct sum of modules|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 ''F''<sup>'''N'''</sup> - see [[Examples of vector spaces#Function spaces|below]].  ''F''<sup>'''N'''</sup> is the ''[[Product (category theory)|product]]'' of countably many copies of ''F''.

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