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Set of functions between two fixed sets
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
Function_space
Mathematical description of quantum state
mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions and form a Hilbert space. The inner product
Wave_function
Function spaces generalizing finite-dimensional p norm spaces
mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes
Lp_space
Normed vector space that is complete
term "Banach space" and Banach in turn then coined the term "Fréchet space". Banach spaces originally grew out of the study of function spaces by Hilbert
Banach_space
Algebraic structure in linear algebra
of topological vector spaces, which include function spaces, inner product spaces, normed spaces, Hilbert spaces and Banach spaces. In this article, vectors
Vector_space
Concept within complex analysis
In complex analysis, the Hardy spaces (or Hardy classes) H p {\displaystyle H^{p}} are spaces of holomorphic functions on the unit disk or upper half
Hardy_space
Vector space of functions in mathematics
mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function together with its derivatives
Sobolev_space
Mapping which preserves all topological properties of a given space
or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are
Homeomorphism
Function space of all functions whose derivatives are rapidly decreasing
Schwartz space S {\displaystyle {\mathcal {S}}} is the function space of all functions whose derivatives of all orders are rapidly decreasing. This space has
Schwartz_space
Element of a basis for a function space
In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a
Basis_function
Type of vector space in math
Euclidean vector spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized
Hilbert_space
Association of one output to each input
mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the
Function_(mathematics)
Function whose squared absolute value has finite integral
square-integrable function, also called a quadratically integrable function or L 2 {\displaystyle L^{2}} function or square-summable function, is a real- or
Square-integrable_function
Topological vector spaces
In mathematical analysis, the spaces of test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application
Spaces of test functions and distributions
Spaces_of_test_functions_and_distributions
Mathematical function with no sudden changes
of functions are real numbers and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The
Continuous_function
Kind of mathematical function
theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of
Measurable_function
Degree of differentiability of a function or map
is local and is therefore first made for functions defined on open subsets of Euclidean space. For functions on closed intervals, closures of open sets
Smoothness
Point to which functions converge in analysis
to another function g(y), which is in the function space T → R . {\displaystyle T\to \mathbb {R} .} The "closeness" in this function space may be measured
Limit_of_a_function
Generalized function whose value is zero everywhere except at zero
Dirac delta function (or δ {\displaystyle {\boldsymbol {\delta }}} distribution), also known as the unit impulse, is a generalized function on the real
Dirac_delta_function
Property of a mathematical space
M-theory (7D hyperspace + 4D), and the state-space of quantum mechanics is an infinite-dimensional function space. The concept of dimension is not restricted
Dimension
Value approached by a mathematical object
the space. Prominent examples of function spaces with some notion of convergence are Lp spaces and Sobolev space. Suppose f is a real-valued function and
Limit_(mathematics)
Type of continuity of a complex-valued function
continuous, if for a real or complex-valued function f {\displaystyle f} on d {\displaystyle d} -dimensional Euclidean space, i.e. f : Ω → R {\displaystyle f:\Omega
Hölder_condition
Mathematical space with a notion of distance
metric space is a set together with a notion of distance between its points. The distance is measured by a function called a metric or distance function. Metric
Metric_space
Collection of random variables
element in a function space. The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set
Stochastic_process
Inputs for which a function's value is non-zero
supported smooth functions on a Euclidean space are sometimes called bump functions. Mollifiers are an important special case of bump functions as they can
Support_(mathematics)
Space of bounded sequences
, the vector space of essentially bounded measurable functions with the essential supremum norm, are two closely related Banach spaces. In fact the former
L-infinity
Mathematics of real numbers and real functions
Lebesgue integration, and function spaces. Real analysis is also known, especially in older books, as the theory of functions of a real variable, in contrast
Real_analysis
Mathematical function of a linear operator
of a linear operator D defined on some function space is any non-zero function f {\displaystyle f} in that space that, when acted upon by D, is only multiplied
Eigenfunction
Mathematical function
of a function of a real variable may be any set. However, it is often assumed to have a structure of R {\displaystyle \mathbb {R} } -vector space over
Function_of_a_real_variable
Mapping involving integration between function spaces
maps a function from its original function space into another function space via integration, where some of the properties of the original function might
Integral_transform
Real function with finite total variation
In mathematical analysis, a function of bounded variation, also known as BV function, is a real-valued function whose total variation is bounded (finite):
Bounded_variation
Type of function
mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval as
Orthogonal_functions
Type of mathematical function
basis for some function space of interest, hence the name. Sums of radial basis functions are typically used to approximate given functions. This approximation
Radial_basis_function
coefficients in F is vector space over F denoted F[x1, x2, ..., xr]. Here r is the number of variables. See main article at Function space, especially the functional
Examples_of_vector_spaces
Machine learning framework
architectures designed to learn maps between infinite-dimensional function spaces. Neural operators represent an extension of traditional artificial
Neural_operators
Area of mathematics
of vector spaces endowed with some kind of limit-related structure (for example, inner product, norm, or topology) and the linear functions defined on
Functional_analysis
Generalization of the inverse function theorem
him and Jürgen Moser, is a generalization of the inverse function theorem on Banach spaces to settings when the required solution mapping for the linearized
Nash–Moser_theorem
In functional analysis, a Hilbert space
Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Specifically, a Hilbert space H {\displaystyle
Reproducing kernel Hilbert space
Reproducing_kernel_Hilbert_space
Curve whose range contains the unit square
function whose domain is the unit interval [0, 1]. In the most general form, the range of such a function may lie in an arbitrary topological space,
Space-filling_curve
Property holding for typical examples
property of a space is a property that holds at "almost all" points of the space, as in the statement, "If f : M → N is a smooth function between smooth
Generic_property
Type of function space
Orlicz space is a type of function space which generalizes Lp spaces. Like L p {\displaystyle L^{p}} spaces, they are Banach spaces. The spaces are named
Orlicz_space
Branch of mathematics
Mathematical analysis is the branch of mathematics that studies functions, spaces, and operators through methods of approximation and convergence. It
Mathematical_analysis
Function which is integrable on its domain
importance of such functions lies in the fact that their function space is similar to p-integrable function spaces ( L p {\textstyle L^{p}} spaces), but its members
Locally_integrable_function
analysis and operator theory, a Bergman space, named after Stefan Bergman, is a function space of holomorphic functions in a domain D of the complex plane
Bergman_space
Mathematical transform that expresses a function of time as a function of frequency
generalized to functions of several variables on Euclidean space, sending a function of 3-dimensional "position space" to a function of 3-dimensional
Fourier_transform
Colour space defined by the CIE in 1931
standard observer is defined by the 3 color matching functions in one of the CIE 1931 color spaces. Due to the design of the experiments, the standard
CIE_1931_color_space
by the space of continuous functions on a compact Hausdorff space X {\displaystyle X} with values in the real or complex numbers. This space, denoted
Space of continuous functions on a compact space
Space_of_continuous_functions_on_a_compact_space
Methods for solving differential equations
derive the basis representation for the function space of our solution u {\displaystyle u} . The function space is defined as S h p := { v ∈ L 2 ( R )
Discontinuous_Galerkin_method
Function space
of a function. The Lorentz space on a measure space ( X , μ ) {\displaystyle (X,\mu )} is the space of extended-complex-valued measurable functions f :
Lorentz_space
Broad concept generalizing scalars in mathematics and physics
of topological vector spaces, which include function spaces, inner product spaces, normed spaces, Hilbert spaces and Banach spaces. Every algebra over a
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Real-valued function
function of bounded mean oscillation, also known as a BMO function, is a real-valued function whose mean oscillation is bounded (finite). The space of
Bounded_mean_oscillation
Objects that generalize functions
reinterprets functions as linear functionals acting on a space of test functions. Standard functions act by integration against a test function, but many
Distribution (mathematical analysis)
Distribution_(mathematical_analysis)
Mapping arbitrary data to fixed-size values
all inputs is some sort of metric space, and the hashing function can be interpreted as a partition of that space into a grid of cells. The table is
Hash_function
Exponential object, category-theoretic equivalent First-class function Function space, set-theoretic equivalent Pierce, Benjamin C. (2002). Types and
Function_type
Linear map or polynomial function of degree one
is a kind of function between vector spaces. In calculus, analytic geometry and related areas, a linear function is a polynomial of degree one or less
Linear_function
Auxiliary functions used to probe equations, distributions, and weak formulations
by any (paracompact) smooth manifold. The space D(U) of test functions on U is defined as follows. A function φ {\displaystyle \varphi } : U → R is said
Test_function
Special mathematical function defined as sin(x)/x
bandlimited signal from uniformly spaced samples of that signal. The sinc filter is used in signal processing. The function itself was first mathematically
Sinc_function
Mode of convergence of a function sequence
real-valued functions, although the concept is readily generalized to functions mapping to metric spaces and, more generally, uniform spaces (see below)
Uniform_convergence
Artificial intelligence method for mathematical discovery
FunSearch (short for searching in the function space) is an artificial intelligence method developed by Google DeepMind for discovering computer programs
FunSearch
Type of topological space
mathematics, Bochner spaces are a generalization of the concept of L p {\displaystyle L^{p}} spaces to functions whose values lie in a Banach space which is not
Bochner_space
Mathematical function that outputs real values
many function spaces consist of real-valued functions. Let F ( X , R ) {\displaystyle {\mathcal {F}}(X,{\mathbb {R} })} be the set of all functions from
Real-valued_function
Set of all things that may be the input of a mathematical function
coordinate space C n . {\displaystyle \mathbb {C} ^{n}.} Sometimes such a domain is used as the domain of a function, although functions may be defined
Domain_of_a_function
Mathematical theorem in real analysis
continuous functions is continuous. More precisely, let X be a topological space, let Y be a metric space, and let ƒn : X → Y be a sequence of functions converging
Uniform_limit_theorem
Concept in theoretical computer science
Turing machine of n states can write on a tape. The function space ( n ) {\displaystyle {\text{space}}(n)} is defined to be the maximal number of tape squares
Busy_beaver
Mathematical set with some added structure
represent numbers, functions on another space, or subspaces of another space. It is the relationships that define the nature of the space. More precisely
Space_(mathematics)
Evaluation of a function on its argument
allows a homotopy deformation to be viewed as a continuous path in the space of functions. Likewise, valid mutations (refactorings) of computer programs can
Function_application
Vector space of infinite sequences
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
Sequence_space
Transforming a function in such a way that it only takes a single argument
is the technique of translating a function that takes multiple arguments into a sequence of families of functions, each taking a single argument. In
Currying
Generalization of Sobolev spaces
space when 1 ≤ p, q ≤ ∞. These spaces, as well as the similarly defined Triebel–Lizorkin spaces, serve to generalize more elementary function spaces such
Besov_space
Type of regression analysis
into a moments problem in a natural function space, usually built around generalizations of the Meijer-G function. By not requiring a priori specification
Symbolic_regression
On when a family of real, continuous functions has a uniformly convergent subsequence
by Fréchet (1906), to sets of real-valued continuous functions with domain a compact metric space (Dunford & Schwartz 1958, p. 382). Modern formulations
Arzelà–Ascoli_theorem
In functional analysis, the Barron space is a function space. It is a Banach space. It originated from the study of universal approximation properties
Barron_space
Joint command of the British Armed Forces
Civil Service. The UKSC has three functions: space operations, space workforce generation, and space capability. UK Space Command was established on 1 April
United_Kingdom_Space_Command
special functions which developed out of statistics and mathematical physics. A modern, abstract point of view contrasts large function spaces, which are
List of mathematical functions
List_of_mathematical_functions
Generalization of perpendicularity
function spaces, families of functions are used to form an orthogonal basis, such as in the contexts of orthogonal polynomials, orthogonal functions,
Orthogonality_(mathematics)
Canadian mathematician
Javad Mashreghi is a mathematician and author working in function space theory, functional analysis and complex analysis. He is a professeur titulaire
Javad_Mashreghi
operator on a certain function space that conserves the mass (the so-called Markov property). If the underlying measurable space is topologically sufficiently
Markov_operator
Mathematical property of sets
Note that in a constructive setting, a countability claim about the function space N N {\displaystyle {\mathbb {N} }^{\mathbb {N} }} out of the full set
Subcountability
Discrete-variable probability distribution
and statistics, a probability mass function (sometimes called probability function or frequency function) is a function that gives the probability that a
Probability_mass_function
Type of mathematical space
real-valued function on a finite set is bounded and attains its maximum and minimum, every continuous real-valued function on a compact space has these
Compact_space
In mathematics, specifically functional analysis, a weighted space is a function space equipped with a weighted norm, which is a finite norm (or semi-norm)
Weighted_space
Function specifying the behavior of a component in an electronic or control system
a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that models
Transfer_function
In mathematics, an operator or transform is a function from one space of functions to another. Operators occur commonly in engineering, physics and mathematics
List_of_mathematic_operators
Moroccan-born American mathematician (born 1959)
fixed point theory, nonlinear functional analysis, metric spaces, and modular function spaces. He is a professor of mathematics at Khalifa University in
Mohamed_Amine_Khamsi
Type of topology
maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory
Compact-open_topology
Technique to make a model more generalizable and transferable
for training. In the case of a general function, the norm of the function in its reproducing kernel Hilbert space is: min f ∑ i = 1 n V ( f ( x ^ i ) ,
Regularization_(mathematics)
Recurrence equation on a function space, that involves integration
mathematics, an integrodifference equation is a recurrence relation on a function space, of the following form: n t + 1 ( x ) = ∫ Ω k ( x , y ) f ( n t ( y
Integrodifference_equation
Type of mathematical convergence in topology
{\mathcal {T}})} be a topological space and ( Y , d Y ) {\displaystyle (Y,d_{Y})} be a metric space. A sequence of functions f n : X → Y {\displaystyle f_{n}:X\to
Compact_convergence
Finite or infinite ordered list of elements
topological space. A sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose
Sequence
Smooth and compactly supported function
smooth functions Non-analytic smooth function – Mathematical functions which are smooth but not analytic Schwartz space – Function space of all functions whose
Bump_function
sequence of positive linear operators on a function space by examining its convergence on a finite set of test functions. The Korovkin approximation is named
Korovkin_approximation
Theorem
\}} be a continuous extended real-valued function. Define a nonlinear functional F {\displaystyle F} on functions u : Ω → R m {\displaystyle u:\Omega \to
Tonelli's theorem (functional analysis)
Tonelli's_theorem_(functional_analysis)
Operation on mathematical functions
kind of multiplication on a function space, but has very different properties from pointwise multiplication of functions (e.g. composition is not commutative)
Function_composition
Mathematical function on a space that is invariant under the action of some group
function is a function on a space that is invariant under the action of some group, in other words a function on the quotient space. Often the space is
Automorphic_function
Mathematical relation assigning a probability event to a cost
optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one
Loss_function
Hilbert space of square-integrable holomorphic functions of n complex variables
Segal–Bargmann space (for Irving Segal and Valentine Bargmann), also known as the Bargmann space or Bargmann–Fock space, is the space of holomorphic functions F in
Segal–Bargmann_space
On the dimension of vector space duals
any function space. The theorem is named after Paul Erdős and Irving Kaplansky. Let E {\displaystyle E} be an infinite-dimensional vector space over
Erdős–Kaplansky_theorem
Mathematical term in complex analysis
family is a pre-compact subset of the space of continuous functions. Informally, this means that the functions in the family are not widely spread out
Normal_family
Class of operator mapping
that maps a space of functions on a topological space to another space of functions on some domain in such a way that the value of the function output at
Nonlocal_operator
Concept in complexity theory
natural function f for which the theorem is true. Time-constructible functions are often used to provide such a definition. Space-constructible functions are
Constructible_function
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