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 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". 
In linear algebra.
Let be a vector space over a field and let be any set. The functions → can be given the structure of a vector space over where the operations are defined pointwise, that is, for any , : → , any in , and any in , define
When the domain has additional structure, one might consider instead the subset (or subspace) of all such functions which respect that structure. For example, if is also a vector space over , the set of linear maps → form a vector space over with pointwise operations (often denoted Hom(,)). One such space is the dual space of : the set of linear functionals → with addition and scalar multiplication defined pointwise.
Examples.
Function spaces appear in various areas of mathematics:
Functional analysis.
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 formula_3
Norm.
If is an element of the function space formula_23 of all continuous functions that are defined on a closed interval [a,b], the norm formula_24 defined on formula_23 is the maximum absolute value of for ,
is called the "uniform norm" or "supremum norm" ('sup norm').