Function spaces

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. Often in mathematical jargon, especially in analysis or geometry, a function could refer to a map of the form X → R {\displaystyle X\to \mathbb {R} } or X → C {\displaystyle X\to \mathbb {C} } where X {\displaystyle X} is the space in question. Whilst other maps of the form X → Y {\displaystyle X\to Y} between any two spaces are simply referred to as maps. Example of this can be the space of compactly supported functions on a topological space. However in a larger context a function space could just consist of a set of functions (set theoretically) equipped with possibly some extra structure.