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 Fn. Likewise, if X is the set of natural numbers, N, then this space is just F∞. A canonical basis for (FX)0 is the set of functions {δx | x ∈ X} defined by The dimension of (FX)0 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 (FX)0. Generalized coordinate space may also be understood as the direct sum of |X| copies of F (i.e. one for each point in X): 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 FX. Linear maps An important example arising in the context of linear algebra itself is the vector space of linear maps. 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 WV since it is closed under addition and scalar multiplication. Note that L(Fn,Fm) can be identified with the space of matrices Fm×n 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 Fm×n. 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 functions from X to R. This is a vector subspace of RX 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 RR if the equation is linear. This is because differentiation is a linear operation, i.e., (a f + b g)′ = a f′ + b g′, where ′ is the differentiation operator. Field extensions Suppose K is a subfield of F (cf. field extension). Then F can be regarded as a vector