# 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 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 `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