# Function space

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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.

## 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

{\displaystyle {\begin{aligned}(f+g)(x)&=f(x)+g(x)\\(c\cdot f)(x)&=c\cdot f(x)\end{aligned}}}

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 a 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. Here we use the real line as an example domain, but the spaces below exist on suitable open subsets ${\displaystyle \Omega \subseteq \mathbf {R} ^{n}}$

• ${\displaystyle C(\mathbf {R} )}$ continuous functions endowed with the uniform norm topology
• ${\displaystyle C_{c}(\mathbf {R} )}$ continuous functions with compact support
• ${\displaystyle B(\mathbf {R} )}$ bounded functions
• ${\displaystyle C_{0}(\mathbf {R} )}$ continuous functions which vanish at infinity
• ${\displaystyle C^{r}(\mathbf {R} )}$ continuous functions that have continuous first r derivatives.
• ${\displaystyle C^{\infty }(\mathbf {R} )}$ smooth functions
• ${\displaystyle C_{c}^{\infty }(\mathbf {R} )}$ smooth functions with compact support
• ${\displaystyle C^{\omega }(\mathbf {R} )}$ real analytic functions
• ${\displaystyle L^{p}(\mathbf {R} )}$, for ${\displaystyle 1\leq p\leq \infty }$, is the Lp space of measurable functions whose p-norm ${\textstyle \|f\|_{p}=\left(\int _{\mathbf {R} }|f|^{p}\right)^{1/p}}$ is finite
• ${\displaystyle {\mathcal {S}}(\mathbf {R} )}$, the Schwartz space of rapidly decreasing smooth functions and its continuous dual, ${\displaystyle {\mathcal {S}}'(\mathbf {R} )}$ tempered distributions
• ${\displaystyle D(\mathbf {R} )}$ compact support in limit topology
• ${\displaystyle W^{k,p}}$ Sobolev space of functions whose weak derivatives up to order k are in ${\displaystyle L^{p}}$
• ${\displaystyle {\mathcal {O}}_{U}}$ holomorphic functions
• linear functions
• piecewise linear functions
• continuous functions, compact open topology
• all functions, space of pointwise convergence
• Hardy space
• Hölder space
• Càdlàg functions, also known as the Skorokhod space
• ${\displaystyle {\text{Lip}}_{0}(\mathbf {R} )}$, the space of all Lipschitz functions on ${\displaystyle \mathbf {R} }$ that vanish at zero.

## Norm

If y is an element of the function space ${\displaystyle {\mathcal {C}}(a,b)}$ of all continuous functions that are defined on a closed interval [a,b], the norm ${\displaystyle \|y\|_{\infty }}$ defined on ${\displaystyle {\mathcal {C}}(a,b)}$ is the maximum absolute value of y (x) for axb, [2]

${\displaystyle \|y\|_{\infty }\equiv \max _{a\leq x\leq b}|y(x)|\qquad {\text{where}}\ \ y\in {\mathcal {C}}(a,b)}$

is called the uniform norm or supremum norm ('sup norm').

## 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.

## Footnotes

1. Fulton, William; Harris, Joe (1991). Representation Theory: A First Course. Springer Science & Business Media. p. 4. ISBN   9780387974958.
2. Gelfand, I. M.; Fomin, S. V. (2000). Silverman, Richard A. (ed.). Calculus of variations (Unabridged repr. ed.). Mineola, New York: Dover Publications. p. 6. ISBN   978-0486414485.

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