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 has 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.
Let F be a field and let X be any set. The functions X → F can be given the structure of a vector space over F where the operations are defined pointwise, that is, for any f, g : X → F, any x in X, and any c in F, define 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 V and also X itself are vector spaces 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 X: the set of linear functionals X → F with addition and scalar multiplication defined pointwise.
The cardinal dimension of a function space with no extra structure can be found by the Erdős–Kaplansky theorem.
Function spaces appear in various areas of mathematics:
Functional analysis is organized around adequate techniques to bring function spaces as topological vector spaces within reach of the ideas that would apply to normed spaces of finite dimension. Here we use the real line as an example domain, but the spaces below exist on suitable open subsets
If y is an element of the function space of all continuous functions that are defined on a closed interval [a, b], the norm defined on is the maximum absolute value of y (x) for a ≤ x ≤ b,[2]
is called the uniform norm or supremum norm ('sup norm').