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In mathematics, a **function space** is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set `X` into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure, hence the name function *space*.

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Let `V` be a vector space over a field `F` and let `X` be any set. The functions `X` → `V` can be given the structure of a vector space over `F` where the operations are defined pointwise, that is, for any `f`, `g` : `X` → `V`, any `x` in `X`, and any `c` in `F`, define

When the domain `X` has additional structure, one might consider instead the subset (or subspace) of all such functions which respect that structure. For example, if `X` is also a vector space over `F`, the set of linear maps `X` → `V` form a vector space over `F` with pointwise operations (often denoted Hom(`X`,`V`)). One such space is the dual space of `V`: the set of linear functionals `V` → `F` with addition and scalar multiplication defined pointwise.

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}.

- As a special case, the power set of a set
- The set of bijections from
*X*to*Y*is denoted . The factorial notation*X*! may be used for permutations of a single set*X*. - 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.
- In the representation theory of finite groups, given two finite-dimensional representations
`V`and`W`of a group`G`, one can form a representation of`G`over the vector space of linear maps Hom(`V`,`W`) called the Hom representation.^{ [1] }

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. Here we use the real line as an example domain, but the spaces below exist on suitable open subsets

- continuous functions endowed with the uniform norm topology
- continuous functions with compact support
- bounded functions
- continuous functions which vanish at infinity
- continuous functions that have continuous first
*r*derivatives. - smooth functions
- smooth functions with compact support
- real analytic functions
- , for , is the L
^{p}space of measurable functions whose*p*-norm is finite - , the Schwartz space of rapidly decreasing smooth functions and its continuous dual, tempered distributions
- compact support in limit topology
- Sobolev space of functions whose weak derivatives up to order
*k*are in - 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
- , the space of all Lipschitz functions on that vanish at zero.

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 *a*≤*x*≤*b*,^{ [2] }

is called the * uniform norm * or *supremum norm* ('sup norm').

- 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.

- ↑ Fulton, William; Harris, Joe (1991).
*Representation Theory: A First Course*. Springer Science & Business Media. p. 4. ISBN 9780387974958. - ↑ 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.

In mathematics, more specifically in functional analysis, a **Banach space** is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

In mathematics, any vector space * has a corresponding ***dual vector space** consisting of all linear forms on *, together with the vector space structure of pointwise addition and scalar multiplication by constants.*

**Functional analysis** is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

In mathematics, a **normed vector space** or **normed space** is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real world. A norm is a real-valued function defined on the vector space that is commonly denoted and has the following properties:

- It is nonnegative, that is for every vector x, one has
- It is positive on nonzero vectors, that is,
- For every vector x, and every scalar one has
- The triangle inequality holds; that is, for every vectors x and y, one has

In mathematics, an **operator** is generally a mapping or function that acts on elements of a space to produce elements of another space. There is no general definition of an *operator*, but the term is often used in place of *function* when the domain is a set of functions or other structured objects. Also, the domain of an operator is often difficult to be explicitly characterized, and may be extended to related objects. See Operator (physics) for other examples.

In mathematics, **weak topology** is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis.

In mathematics, the ** L^{p} spaces** are function spaces defined using a natural generalization of the

**Distributions**, also known as **Schwartz distributions** or **generalized functions**, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta function.

In the area of mathematics known as functional analysis, a **reflexive space** is a locally convex topological vector space (TVS) such that the canonical evaluation map from *X* into its bidual is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space *X* is reflexive if and only if the canonical evaluation map from *X* into its bidual is surjective; in this case the normed space is necessarily also a Banach space. Note that in 1951, R. C. James discovered a *non*-reflexive Banach space that is isometrically isomorphic to its bidual.

In functional analysis and related areas of mathematics, **Fréchet spaces**, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically *not* Banach spaces.

In mathematics, the **Gelfand representation** in functional analysis has two related meanings:

In functional analysis and related areas of mathematics, **locally convex topological vector spaces** (**LCTVS**) or **locally convex spaces** are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

In mathematics, a **Radon measure**, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space *X* that is finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets. These conditions guarantee that the measure is "compatible" with the topology of the space, and most measures used in mathematical analysis and in number theory are indeed Radon measures.

In mathematics, a **norm** is a function from a real or complex vector space to the nonnegative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance of a vector from the origin is a norm, called the Euclidean norm, or 2-norm, which may also be defined as the square root of the inner product of a vector with itself.

In functional analysis and related areas of mathematics, a **sequence space** is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field *K* of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in *K*, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.

In mathematics, a **nuclear space** is a topological vector space with many of the good properties of finite-dimensional vector spaces. The topology on them can be defined by a family of seminorms whose unit balls decrease rapidly in size. Vector spaces whose elements are "smooth" in some sense tend to be nuclear spaces; a typical example of a nuclear space is the set of smooth functions on a compact manifold.

In mathematics, more particularly in functional analysis, differential topology, and geometric measure theory, a ** k-current** in the sense of Georges de Rham is a functional on the space of compactly supported differential

In functional analysis, the concept of a **compact operator on Hilbert space** is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators in the topology induced by the operator norm. As such, results from matrix theory can sometimes be extended to compact operators using similar arguments. By contrast, the study of general operators on infinite-dimensional spaces often requires a genuinely different approach.

In mathematics, the **Banach–Stone theorem** is a classical result in the theory of continuous functions on topological spaces, named after the mathematicians Stefan Banach and Marshall Stone.

In mathematics, the **injective tensor product** of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so its completion is called the *completed injective tensor products*. Injective tensor products have applications outside of nuclear spaces. In particular, as described below, up to TVS-isomorphism, many TVSs that are defined for real or complex valued functions, for instance, the Schwartz space or the space of continuously differentiable functions, can be immediately extended to functions valued in a Hausdorff locally convex TVS Y with*out* any need to extend definitions from real/complex-valued functions to Y-valued functions.

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