Editing Examples of vector spaces (section)

==Function spaces==
:''See main article at [[Function space]], especially the functional analysis section.''
Let ''X'' be a non-empty arbitrary set and ''V'' an arbitrary vector space over ''F''. The space of all [[function (mathematics)|function]]s from ''X'' to ''V'' is a vector space over ''F'' under [[pointwise]] addition and multiplication. That is, let ''f'' : ''X'' → ''V'' and ''g'' : ''X'' → ''V'' denote two functions, and let ''α'' in ''F''. We define
:<math>(f + g)(x) = f(x) + g(x) </math>
:<math>(\alpha f)(x) = \alpha f(x) </math>
where the operations on the right hand side are those in ''V''. The zero vector is given by the constant function sending everything to the zero vector in ''V''. The space of all functions from ''X'' to ''V'' is commonly denoted ''V''<sup>''X''</sup>.

If ''X'' is finite and ''V'' is finite-dimensional then ''V''<sup>''X''</sup> has dimension |''X''|(dim ''V''), otherwise the space is infinite-dimensional (uncountably so if ''X'' is infinite).

Many of the vector spaces that arise in mathematics are subspaces of some function space. We give some further examples.

===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>.

===Linear maps===
An important example arising in the context of [[linear algebra]] itself is the vector space of [[linear map]]s. Let ''L''(''V'',''W'') denote the set of all linear maps from ''V'' to ''W'' (both of which are vector spaces over ''F''). Then ''L''(''V'',''W'') is a subspace of ''W''<sup>''V''</sup> since it is closed under addition and scalar multiplication.

Note that L(''F''<sup>''n''</sup>,''F''<sup>''m''</sup>) can be identified with the space of matrices ''F''<sup>''m''×''n''</sup> in a natural way.  In fact, by choosing appropriate bases for finite-dimensional spaces V and W, L(V,W) can also be identified with ''F''<sup>''m''×''n''</sup>.  This identification normally depends on the choice of basis.

===Continuous functions===
If ''X'' is some [[topological space]], such as the [[unit interval]] [0,1], we can consider the space of all [[Continuous function (topology)|continuous function]]s from ''X'' to '''R'''. This is a vector subspace of '''R'''<sup>''X''</sup>  since the sum of any two continuous functions is continuous and scalar multiplication is continuous.

===Differential equations===
The subset of the space of all functions from '''R''' to '''R''' consisting of (sufficiently differentiable) functions that satisfy a certain [[differential equation]] is a subspace of '''R'''<sup>'''R'''</sup> if the equation is linear. This is because [[derivative|differentiation]] is a linear operation, i.e., (''a'' ''f'' + ''b'' ''g'')′ = ''a'' ''f''{{space|hair}}′ + ''b'' ''g''′, where ′ is the differentiation operator.