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.

Examples

Function spaces appear in various areas of mathematics:

• In set theory, the set of functions from X to Y may be denoted X → Y or Y^X.
• 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 2^X.
• The set of bijections from X to Y is denoted X ↔ Y. 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.);
• In functional analysis the same is seen for continuous linear transformations, including topologies on the vector spaces in the above, and many of the major examples are function spaces carrying a topology; the best known examples include Hilbert spaces and Banach spaces.
• In functional analysis the set of all functions from the natural numbers to some set X is called a sequence space. It consists of the set of all possible sequences of elements of X.
• In topology, one may attempt to put a topology on the space of continuous functions from a topological space X to another one Y, with utility depending on the nature of the spaces. A commonly used example is the compact-open topology, e.g. loop space. Also available is the product topology on the space of set theoretic functions (i.e. not necessarily continuous functions) Y^X. In this context, this topology is also referred to as the topology of pointwise convergence.
• In algebraic topology, the study of homotopy theory is essentially that of discrete invariants of function spaces;
• In the theory of stochastic processes, the basic technical problem is how to construct a probability measure on a function space of paths of the process (functions of time);
• In category theory the function space is called an exponential object or map object. It appears in one way as the representation canonical bifunctor; but as (single) functor, of type [X, -], it appears as an adjoint functor to a functor of type (-×X) on objects;
• In functional programming and lambda calculus, function types are used to express the idea of higher-order functions.
• In domain theory, the basic idea is to find constructions from partial orders that can model lambda calculus, by creating a well-behaved cartesian closed category.

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.

• Schwartz space of smooth functions of rapid decrease and its dual, tempered distributions
• Lp space
• κ(R) continuous functions with compact support endowed with the uniform norm topology
• B(R) bounded functions
• C₀(R) continuous functions which vanish at infinity
• C^r(R) continuous functions that have continuous first r derivatives.
• C^∞(R) Smooth functions
• C^∞_c smooth functions with compact support
• D(R) compact support in limit topology
• W^k,p Sobolev space
• O_U holomorphic functions
• linear functions
• piecewise linear functions
• continuous functions, compact open topology
• all functions, space of pointwise convergence
• Hardy space
• Hölder space
• Càdlàg functions, also known as the Skorokhod space

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 a ≤ x ≤ b,^[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

• List of mathematical functions
• Linear algebra
• Vector space
• Banach space
• Hilbert space
• Clifford algebra
• Tensor field
• Spectral theory

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.