Editing Examples of vector spaces (section)

===Generalized coordinate space===
Let ''X'' be an arbitrary set. Consider the space of all functions from ''X'' to ''F'' which vanish on all but a finite number of points in ''X''. This space is a vector subspace of ''F''<sup>''X''</sup>, the space of all possible functions from ''X'' to ''F''. To see this, note that the union of two finite sets is finite, so that the sum of two functions in this space will still vanish outside a finite set.

The space described above is commonly denoted (''F''<sup>''X''</sup>)<sub>0</sub> and is called ''generalized coordinate space'' for the following reason. If ''X'' is the set of numbers between 1 and ''n'' then this space is easily seen to be equivalent to the coordinate space ''F''<sup>''n''</sup>. Likewise, if ''X'' is the set of [[natural number]]s, '''N''', then this space is just ''F''<sup>∞</sup>.

A canonical basis for (''F''<sup>''X''</sup>)<sub>0</sub> is the set of functions {δ<sub>''x''</sub> | ''x'' ∈ ''X''} defined by
:<math>\delta_x(y) = \begin{cases}1 \quad x = y \\ 0 \quad x \neq y\end{cases}</math>
The dimension of (''F''<sup>''X''</sup>)<sub>0</sub> is therefore equal to the [[cardinality]] of ''X''. In this manner we can construct a vector space of any dimension over any field. Furthermore, ''every vector space is isomorphic to one of this form''. Any choice of basis determines an isomorphism by sending the basis onto the canonical one for (''F''<sup>''X''</sup>)<sub>0</sub>.

Generalized coordinate space may also be understood as the [[direct sum of modules|direct sum]] of |''X''| copies of ''F'' (i.e. one for each point in ''X''):
:<math>(\mathbf F^X)_0 = \bigoplus_{x\in X}\mathbf F.</math>
The finiteness condition is built into the definition of the direct sum. Contrast this with the [[direct product]] of |''X''| copies of ''F'' which would give the full function space ''F''<sup>''X''</sup>.