# 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 XV can be given the structure of a vector space over F where the operations are defined pointwise, that is, for any f, g : XV, 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 X is also vector space over F, the set of linear maps XV 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 VF 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.

## Norm

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]

## Bibliography

• Kolmogorov, A. N., & Fomin, S. V. (1967). Elements of the theory of functions and functional analysis. Courier Dover Publications.
• Stein, Elias; Shakarchi, R. (2011). Functional Analysis: An Introduction to Further Topics in Analysis. Princeton University Press.