Function space

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.

Function spaces appear in various areas of mathematics:

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.

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

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