Function space

In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space have a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure, hence the name function space.

In linear algebra

Let V be a vector space over a field F and let X be any set. The functions X → V can be given the structure of a vector space over F where the operations are defined pointwise, that is, for any f, g : X → V, any x in X, and any c in F, define

{\begin{aligned}(f+g)(x)&=f(x)+g(x)\\(c\cdot f)(x)&=c\cdot f(x)\end{aligned}}

When the domain X has additional structure, one might consider instead the subset (or subspace) of all such functions which respect that structure. For example, if X is also vector space over F, the set of linear maps X → V form a vector space over F with pointwise operations (often denoted Hom(X,V)). One such space is the dual space of V: the set of linear functionals V → F with addition and scalar multiplication defined pointwise.