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.


Function spaces appear in various areas of mathematics:

  • In set theory, the set of functions from X to Y may be denoted XY or YX.
  • As a special case, the power set of a set X may be identified with the set of all functions from X to {0, 1}, denoted 2X.
  • The set of bijections from X to Y is denoted XY. The factorial notation X! may be used for permutations of a single set X.
  • In linear algebra the set of all linear transformations from a vector space V to another one, W, over the same field, is itself a vector space (with the natural definitions of 'addition of functions' and 'multiplication of functions by scalars' : this vector space is also over the same field as that of V and W.);

Functional analysis[edit]

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.


  • 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.

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