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[edit]

Function spaces appear in various areas of mathematics:

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.

Norm[edit]

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 axb,[1]

Bibliography[edit]

  • 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.

See also[edit]

Footnotes[edit]

  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.