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:

  • 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.

Norm[edit]

If y is an element of the function space  \mathcal {C}(a,b) of all continuous functions that are defined on a closed interval [a,b], the norm || y || defined on  \mathcal {C}(a,b) is the maximum absolute value of y (x) for axb,[1]

 \| y \| \equiv \max_{a \le x \le b} |y(x)| \qquad \text{where} \ \ y \in \mathcal {C}(a,b) \, .

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.