# Function space

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In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space (including metric spaces), a vector space, or both. Namely, if Y is a field, functions have inherent vector structure with two operations of pointwise addition and multiplication to a scalar. Topological and metrical structures of function spaces are more diverse.

## 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 ${\displaystyle {\mathcal {C}}(a,b)}$ 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,[1]

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

## 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. ^ 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.