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:

  • 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

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  \mathcal {C}(a,b) of all continuous functions that are defined on a closed interval [a,b], the norm \|y\|_\infty 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

  • 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

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.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>