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 or a vector space or both. Function spaces appear in various areas of mathematics: Euclid, detail from The School of Athens by Raphael. ... Partial plot of a function f. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...

• in set theory, the power set of a set X may be identified with the set of all functions from X to {0,1};, denoted 2^X. More generally, the set of functions X → Y is denoted Y^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;

• 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. 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. 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 lambda calculus and functional programming, function space types are used to express the idea of higher-order function.

• 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.

Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ... In mathematics, a topological vector space X is a real or complex vector space which is endowed with a Hausdorff topology such that vector addition X × X → X and scalar multiplication K × X → X are continuous (where the product topologies are used and the base field K carries its standard... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ... A natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ... In functional analysis and related areas of mathematics, a sequence space is an important class of function space. ... This is a page about mathematics. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ... Suppose { fn } is a sequence of functions sharing the same domain in common (for the moment, we defer making precise the nature of the values of these functions, but the reader may take them to be real numbers). ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... In the mathematics of probability, a stochastic process is a random function. ... In mathematics, a probability space is a set S, together with a σ-algebra X on S and a measure P on that σ-algebra such that P(S) = 1. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, specifically in category theory, an exponential object is the categorical equivalent of a function space in set theory. ... The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ... The lambda calculus is a formal system designed to investigate function definition, function application, and recursion. ... Functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions. ... In mathematics and computer science, higher-order functions are functions which can take other functions as arguments, and may also return functions as results. ... Domain theory is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains. ... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... In category theory, a category is cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. ...

List of function spaces

Functional analysis

Abstract spaces

• Locally convex space: a vector space with a collection of seminorms (equivalently, a local base of convex sets).
• Fréchet space: a vector space with a countable collection of seminorms (equivalently, a translation invariant metric).
• Banach space: a vector space with a finite collection of seminorms (equivalently, a single norm).
• Hilbert space: a vector space with an inner product.

In functional analysis, a topological vector space is called locally convex if its topology is defined by a set of convex neighborhoods of 0. ... In functional analysis, a seminorm is a function on a vector space with certain properties characteristic of a measure of length. A space with such a seminorm is then known as a seminormed space. ... This article deals with Fréchet spaces in functional analysis. ... In mathematics the term countable set is used to describe the size of a set, e. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... The word norm coming from the latin word norma which means angle measure or (lawlike) rule, has a number of meanings: A social or sociological norm; see norm (sociology). ... In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ... // Definition Inner Product of two vectors Given twoN-by-1 column vectors v and u, the inner product is defined as the scalar quantity α resulting from where or equivalently indicates the conjugate transpose operator applied to vector v. ...

Concrete spaces

• Schwartz space of smooth functions of rapid decrease and its dual, tempered distributions
• Lp space
• κ(R) continuous compact support with uniform norm
• B(R) bounded continuous (Bounded function)
• C_∞(R) functions which vanish at infinity
• C^∞(R) Smooth_functions
• C^∞₀ smooth compact support uniform norm (and the one with derivatives)
• 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

In mathematics, Schwartz space is the function space of rapidly decreasing functions. ... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... In mathematics, a Sobolev space is a normed space of functions obtained by imposing on a function f and its derivatives up to some order k the condition of finite Lp norm, for given p ≥ 1. ... In complex analysis, the Hardy spaces are analogues of the Lp spaces of functional analysis. ... A real-valued function f on a metric space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants C, α, such that, , . This condition obviously generalises to functions between any two metric spaces. ...

See also

• Sequence space