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Whose values lie in an infinite-dimensional vector space
An infinite-dimensional vector function is a function whose values lie in an infinite-dimensional topological vector space, such as a Hilbert space or
Infinite-dimensional vector function
Infinite-dimensional_vector_function
Number of vectors in any basis of the vector space
finite-dimensional if the dimension of V {\displaystyle V} is finite, and infinite-dimensional if its dimension is infinite. The dimension of the vector space
Dimension_(vector_space)
Function valued in a vector space; typically a real or complex one
multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could be a scalar or a vector (that is, the dimension of the domain
Vector-valued_function
Algebraic structure in linear algebra
the vector spaces are isomorphic). A vector space is finite-dimensional if its dimension is a natural number. Otherwise, it is infinite-dimensional, and
Vector_space
Broad concept generalizing scalars in mathematics and physics
its dimension is an infinite cardinal. Finite-dimensional vector spaces occur naturally in geometry and related areas. Infinite-dimensional vector spaces
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Holomorphic functions in infinite dimensions
mathematics, infinite-dimensional holomorphy is a branch of functional analysis. It is concerned with generalizations of the concept of holomorphic function to
Infinite-dimensional holomorphy
Infinite-dimensional_holomorphy
Group that is also a differentiable manifold with group operations that are smooth
In M-theory, for example, a 10-dimensional SU(N) gauge theory becomes an 11-dimensional theory when N becomes infinite. Adjoint representation of a Lie
Lie_group
Area of mathematics
of vector spaces endowed with a topology, in particular infinite-dimensional spaces. In contrast, linear algebra deals mostly with finite-dimensional spaces
Functional_analysis
Normed vector space that is complete
{\displaystyle L^{p}} space – Function spaces generalizing finite-dimensional p norm spaces Sobolev space – Vector space of functions in mathematics Banach lattice –
Banach_space
Concept in linear algebra
of a coordinate vector can also be used for infinite-dimensional vector spaces, as addressed below. Let V be a vector space of dimension n over a field
Coordinate_vector
Element of a basis for a function space
of basis functions. In finite-dimensional vector spaces this representation is purely algebraic and involves only finitely many basis functions, whereas
Basis_function
Length in a vector space
also refer to a norm that can take infinite values or to certain functions parametrised by a directed set. Given a vector space X {\displaystyle X} over a
Norm_(mathematics)
Mathematical description of quantum state
spaces originally refer to infinite dimensional complete inner product spaces they, by definition, include finite dimensional complete inner product spaces
Wave_function
Set of vectors used to define coordinates
with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. Basis vectors find applications
Basis_(linear_algebra)
Smooth approximation of one-hot arg max
probably highly-dimensional, input to vectors in a K-dimensional space R K {\displaystyle \mathbb {R} ^{K}} . The standard softmax function is often used
Softmax_function
Notation for quantum states
and linear operators on complex vector spaces together with their dual spaces both in the finite- and infinite-dimensional cases. It is specifically designed
Bra–ket_notation
Derivative defined on normed spaces
Infinite-dimensional holomorphy – Holomorphic functions in infinite dimensions Infinite-dimensional vector function – Whose values lie in an infinite-dimensional
Fréchet_derivative
Type of vector space in math
two-dimensional Euclidean plane and three-dimensional space to spaces of any finite or infinite dimension. A Hilbert space is an abstract vector space
Hilbert_space
Vector space on which a distance is defined
same vector space are called equivalent if they define the same topology. On a finite-dimensional vector space (but not infinite-dimensional vector spaces)
Normed_vector_space
number or a vector, but rather a continuous quantity, for example a function or the shape of a body. Such a problem is an infinite-dimensional optimization
Infinite-dimensional optimization
Infinite-dimensional_optimization
Property of a mathematical space
mechanics is an infinite-dimensional function space. The concept of dimension is not restricted to physical objects. High-dimensional spaces frequently
Dimension
Mathematical idealization of the trace left by a moving point
Path (topology) Polygonal curve Position vector Vector-valued function Infinite-dimensional vector function Winding number In current mathematical usage
Curve
In mathematics, vector space of linear forms
mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically
Dual_space
Vector space with a notion of nearness
of Montel spaces. An infinite-dimensional Montel space is never normable. The existence of a norm for a given topological vector space is characterized
Topological_vector_space
Mathematical operation on vectors in 3D space
significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E {\displaystyle E} ), and is denoted
Cross_product
Geometric model of the physical space
rarely, tri-dimensional space. Most commonly, it means the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which
Three-dimensional_space
Generalization of the one-dimensional normal distribution to higher dimensions
generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate
Multivariate normal distribution
Multivariate_normal_distribution
Set of methods for supervised statistical learning
stability. More formally, a support vector machine constructs a hyperplane or set of hyperplanes in a high or infinite-dimensional space, which can be used for
Support_vector_machine
Mathematical function, in linear algebra
linear mapping) is a particular kind of function between vector spaces, which respects the basic operations of vector addition and scalar multiplication.
Linear_map
Function that is continuous everywhere but differentiable nowhere
motion necessitated infinitely jagged functions (nowadays known as fractal curves). In Weierstrass's original paper, the function was defined as a Fourier
Weierstrass_function
Generalization of the concept of directional derivative
construction of differential calculus Infinite-dimensional vector function – Whose values lie in an infinite-dimensional vector space Quasi-derivative – Generalization
Gateaux_derivative
Kind of mathematical function
measurable functions as exclusively real-valued ones with respect to the Borel algebra. If the values of the function lie in an infinite-dimensional vector space
Measurable_function
Mathematical concept
of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces
Infinity
Mathematical folklore
In mathematics, an infinite-dimensional Lebesgue measure is a measure defined on infinite-dimensional normed vector spaces, such as Banach spaces, which
Infinite-dimensional Lebesgue measure
Infinite-dimensional_Lebesgue_measure
Vector representing the position of a point with respect to a fixed origin
vector is used mostly in the fields of differential geometry, mechanics and occasionally vector calculus. Frequently this is used in two-dimensional or
Position_(geometry)
sees that a vector space need not be isomorphic to its double dual if it is infinite dimensional, in contrast to the finite dimensional case. Starting
Examples_of_vector_spaces
Vector field that is the gradient of some function
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property
Conservative_vector_field
Mathematical parametrization of vector spaces by another space
a finite-dimensional real vector space and hence has a dimension k x {\displaystyle k_{x}} . The local trivializations show that the function x → k x {\displaystyle
Vector_bundle
Generalized function whose value is zero everywhere except at zero
\cos(px-p\alpha )\ .} Later, an infinitesimal formula for an infinitely tall, unit impulse delta function (infinitesimal version of Cauchy distribution) explicitly
Dirac_delta_function
Concepts from linear algebra
not exist. The converse is true for finite-dimensional vector spaces, but not for infinite-dimensional vector spaces. In general, the operator (T − λI)
Eigenvalues_and_eigenvectors
Fundamental object of geometry
As zero-dimensional objects, points are usually taken to be the fundamental indivisible elements comprising the space, of which one-dimensional curves
Point_(geometry)
Subject in mathematics
topological vector spaces refers to the extension of measure theory to topological vector spaces. Such spaces are often infinite-dimensional, but many results
Measure theory in topological vector spaces
Measure_theory_in_topological_vector_spaces
Euclidean space without distance and angles
without any size or shape: zero-dimensional. Through any pair of points an infinite straight line can be drawn, a one-dimensional set of points; through any
Affine_space
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
Topological vector spaces
test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application of distributions. Test functions are
Spaces of test functions and distributions
Spaces_of_test_functions_and_distributions
Class of algorithms for pattern analysis
similarity function over all pairs of data points computed using inner products. The feature map in kernel machines is infinite dimensional but only requires
Kernel_method
describes a canonical example. Infinite-dimensional vector function#Crinkled arcs – Whose values lie in an infinite-dimensional vector space Halmos, Paul R. (8
Crinkled_arc
Mathematics of convex functions and sets
formulated in elementary geometric terms. In infinite-dimensional spaces, the topology of the underlying vector space becomes part of the theory. The relevant
Convex_analysis
Operator in differential topology
operation and turns the set of all smooth vector fields on the manifold M {\displaystyle M} into an (infinite-dimensional) Lie algebra. The Lie bracket plays
Lie_bracket_of_vector_fields
Operation in mathematical calculus
and the three-dimensional vector calculus, the wedge product and the calculus of differential forms makes sense in arbitrary dimension and on more general
Integral
Fourier transform of the probability density function
functions can be defined for vector- or matrix-valued random variables, and can also be extended to more generic cases. The characteristic function always
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Type of function
g} . As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space. Conceptually, the
Orthogonal_functions
Vectors whose linear combinations are nonzero
vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence
Linear_independence
Fourier transform of a real-space lattice, important in solid-state physics
results in the same reciprocal lattice.) For an infinite two-dimensional lattice, defined by its primitive vectors ( a 1 , a 2 ) {\displaystyle \left(\mathbf
Reciprocal_lattice
Real-valued number of spatial dimensions
be one-dimensional, but too simple to be two-dimensional. Therefore, its dimension might best be described not by its usual topological dimension of 1 but
Fractal_dimension
generated by real vector spaces endowed with an indefinite quadratic form. Some authors restrict this to the infinite-dimensional case. The universal
Universal_geometric_algebra
Differentiable function in functional analysis
arbitrary topological vector spaces (TVSs) in multiple ways. But when the domain of a TVS-valued function is a subset of a finite-dimensional Euclidean space
Differentiable vector-valued functions from Euclidean space
Differentiable_vector-valued_functions_from_Euclidean_space
Degree of differentiability of a function or map
function as a map between real vector spaces. This should be distinguished from complex differentiability: a complex function that is complex differentiable
Smoothness
Fundamental theorem in condensed matter physics
which are infinite, 1-dimensional and abelian. All irreducible representations of abelian groups are one dimensional. Given they are one dimensional the matrix
Bloch's_theorem
Result about when a matrix can be diagonalized
straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general, the spectral
Spectral_theorem
Mathematical operation on vector spaces
. If V and W are vector spaces of finite dimension, then V ⊗ W {\displaystyle V\otimes W} is finite-dimensional, and its dimension is the product of
Tensor_product
Line or vector perpendicular to a curve or a surface
line, ray, or vector) that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the infinite straight line
Normal_(geometry)
Theorems generalizing the Brouwer fixed-point theorem
In mathematics, a number of fixed-point theorems in infinite-dimensional spaces generalise the Brouwer fixed-point theorem. They have applications, for
Fixed-point theorems in infinite-dimensional spaces
Fixed-point_theorems_in_infinite-dimensional_spaces
Infinite-dimensional vector function – Whose values lie in an infinite-dimensional vector space Hamilton, R. S. (1982). "The inverse function theorem of Nash
Differentiation in Fréchet spaces
Differentiation_in_Fréchet_spaces
Algebraic structure
functions on a vector space V over a field k gives a coordinate-free analog of a polynomial ring. It is denoted by k[V]. If V is finite dimensional and
Ring_of_polynomial_functions
Theorem in mathematics
applies in infinite-dimensional (Banach space) settings, this proof generalizes immediately to the infinite-dimensional version of the inverse function theorem
Inverse_function_theorem
Artificial neural network node function
Polyharmonic splines where c {\displaystyle \mathbf {c} } is the vector representing the function center and a {\displaystyle a} and σ {\displaystyle \sigma
Activation_function
Extension of the factorial function
of the zeta function to a meromorphic function in the complex plane and leads to an immediate proof that the zeta function has infinitely many so-called
Gamma_function
Notion in supervised machine learning
increasing function of its input, such as the sign function or the sigmoid function. This function is called the activation function. The VC dimension of a
Vapnik–Chervonenkis_dimension
Geometric model of the planar projection of the physical universe
two-dimensional because every point in the plane can be described by a linear combination of two independent vectors. The dot product of two vectors A =
Euclidean_plane
Theorem on extension of bounded linear functionals
has the separation property. In 1992, Kakol proved that any infinite dimensional vector space X, there exist TVS-topologies on X that do not have the
Hahn–Banach_theorem
Function acting on function spaces
analysis (so called because various classes of functions form interesting examples of infinite-dimensional vector spaces). The space of sequences of real numbers
Operator_(mathematics)
Special functions of several complex variables
\mathbb {C} ^{n}} is an n-dimensional complex vector, and the superscript T denotes the transpose. The Jacobi theta function is then a special case, with
Theta_function
Topological space that locally resembles Euclidean space
homeomorphic to an open subset of n {\displaystyle n} -dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not self-crossing
Manifold
Fundamental space of geometry
the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are
Euclidean_space
Algebra associated to any vector space
smooth functions in k {\displaystyle k} variables. The two-dimensional Euclidean vector space R 2 {\displaystyle \mathbf {R} ^{2}} is a real vector space
Exterior_algebra
Group of transformations under which the object is invariant
scalar field, a function of position with values in a set of colors or substances; as a vector field; or as a more general function on the object.) The
Symmetry_group
Choice of reference for distinguishing an object and its mirror image
zero-dimensional case. A zero-dimensional vector space has only a single point, the zero vector. Consequently, the only basis of a zero-dimensional vector
Orientation_(vector_space)
Real function with secant line between points above the graph itself
convex function on an open set has no more than one minimum. Even in infinite-dimensional spaces, under suitable additional hypotheses, convex functions continue
Convex_function
Computer graphics images defined by points, lines and curves
two- or three-dimensional cartesian coordinate system, as p = (x, y) or p = (x, y, z). Because almost all shapes consist of an infinite number of points
Vector_graphics
Branch of mathematics that studies abstract algebraic structures
vector space on which the algebraic object is represented. The most important distinction is between finite-dimensional representations and infinite-dimensional
Representation_theory
Tangent spaces of a manifold
{\displaystyle M} . Each tangent space of an n-dimensional manifold is an n-dimensional vector space. If U {\displaystyle U} is an open contractible subset of M
Tangent_bundle
Description of continuous random distribution
density functions in the simple case of a function of a set of two variables. Let us call R → {\displaystyle {\vec {R}}} a 2-dimensional random vector of coordinates
Probability_density_function
Type of signal in signal processing
distributions. Analogous to the case for finite-dimensional random vectors, a probability law on the infinite-dimensional space S ′ ( R ) {\displaystyle {\mathcal
White_noise
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 momentum
Fourier_transform
Axiom of set theory
field extension has a transcendence basis. Every infinite-dimensional vector space contains an infinite linearly independent subset. (In ZF, this statement
Axiom_of_choice
Smooth manifold with an inner product on each tangent space
extended, to a certain degree, to infinite-dimensional manifolds; that is, manifolds that are modeled after a topological vector space; for example, Fréchet
Riemannian_manifold
Description of a quantum-mechanical system
square-integrable functions L 2 {\displaystyle L^{2}} , while the Hilbert space for the spin of a single proton is the two-dimensional complex vector space C 2
Schrödinger_equation
Generalized version of classical Green's function
above equations are 3N × 3N square matrices and u and f are 3N-dimensional column vectors, where N is the total number of atoms in the lattice. The matrix
Multiscale_Green's_function
Theorem in topology
See fixed-point theorems in infinite-dimensional spaces for a discussion of these theorems. There is also finite-dimensional generalization to a larger
Brouwer_fixed-point_theorem
Machine learning kernel function
Typically, these take the form of a function z that maps a single vector to a vector of higher dimensionality, approximating the kernel: ⟨ z ( x )
Radial_basis_function_kernel
Mathematical inequality relating inner products and norms
vectors can describe finite sums (via finite-dimensional vector spaces), infinite series (via vectors in sequence spaces), and integrals (via vectors
Cauchy–Schwarz_inequality
Ornstein–Uhlenbeck operator is a generalization of the Laplace operator to an infinite-dimensional setting. The Ornstein–Uhlenbeck operator plays a significant role
Ornstein–Uhlenbeck_operator
Probability distribution
of the categorical distribution and multinomial distribution. The infinite-dimensional generalization of the Dirichlet distribution is the Dirichlet process
Dirichlet_distribution
Computational tool
makes Schauder bases more suitable for the analysis of infinite-dimensional topological vector spaces including Banach spaces. Schauder bases were described
Schauder_basis
Branch of mathematics
definition of a vector space was introduced by Peano in 1888; by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged
Linear_algebra
In algebra, integer associated to a module
the dimension of a vector space which measures its size. page 153 It is defined to be the length of the longest chain of submodules. For vector spaces
Length_of_a_module
Space of bounded sequences
finite vector, and the consumption set is a vector space with a finite dimension. But in reality, the number of different commodities may be infinite. For
L-infinity
Branch of mathematics studying functions of a complex variable
holomorphic functions that are the differentiable functions of a complex variable. By contrast with the real case, a holomorphic function is always infinitely differentiable
Complex_analysis
Special mathematical function defined as sin(x)/x
absolute maximum at ξ0 = (0, 1). The normalized sinc function has a simple representation as the infinite product: sin ( π x ) π x = ∏ n = 1 ∞ ( 1 − x 2
Sinc_function
INFINITE DIMENSIONAL-VECTOR-FUNCTION
INFINITE DIMENSIONAL-VECTOR-FUNCTION
Male
Greek
(á¼ÎºÏ„ωÏ) Variant spelling of Greek Hektor, EKTOR means "defend; hold fast."
Boy/Male
Spanish
Victor.
Boy/Male
Indian
Infinite.
Male
Scandinavian
 Scandinavian form of Roman Latin Victor, VIKTOR means "conqueror." Compare with another form of Viktor.
Girl/Female
Indian, Telugu
Uni-dimensional
Girl/Female
Tamil
Trikaya | தà¯à®°à®¿à®•à®¾à®¯à®¾
Three dimensional
Trikaya | தà¯à®°à®¿à®•à®¾à®¯à®¾
Boy/Male
American, British, Christian, Danish, Dutch, English, Finnish, French, German, Greek, Hindu, Indian, Irish, Jamaican, Latin, Romanian, Slovenia, Spanish, Swedish, Swiss, Tamil, Ukrainian
Victorious; Conqueror; Winner; Champion; One who Conquers; Victory
Boy/Male
Tamil
Dimensions
Male
Russian
(Cyrillic Виктор): Slavic form of Roman Latin Victor, VIKTOR means "conqueror." In use by the Bulgarians, Russians and Serbians. Compare with another form of Viktor.
Girl/Female
Hindu
Three dimensional
Boy/Male
Hindu, Indian
Dimensions
Girl/Female
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Telugu
Three Dimentional
Boy/Male
Hindi
Infinite.
Male
English
Roman Latin name VICTOR means "conqueror."Â
Boy/Male
English American
Doctor; teacher.
Male
English
 Anglicized form of Scottish Gaelic Eachann, HECTOR means "brown horse." Compare with another form of Hector.
Male
Portuguese
Galician-Portuguese form of Roman Latin Victor, VITOR means "conqueror."
Male
Portuguese
Portuguese form of Latin Hector, HEITOR means "defend; hold fast."
Male
Arthurian
, sir Hector de Maris; (defender).
Boy/Male
Australian, Basque, Czech, Czechoslovakian, Danish, Finnish, French, German, Hungarian, Latin, Polish, Slovenia, Swedish, Swiss, Ukrainian
The Conqueror; Victory; Victorious; Conquer
INFINITE DIMENSIONAL-VECTOR-FUNCTION
INFINITE DIMENSIONAL-VECTOR-FUNCTION
Boy/Male
Christian & English(British/American/Australian)
A Faithful, Loyal Man
Girl/Female
English American Latin
Introduced to Britian by the Hanoverians in the early 18th century, became popular until the...
Male
Finnish
Finnish form of Latin Martinus, MARTTI means "of/like Mars."
Boy/Male
Arabic, Australian, Muslim, Sindhi
One who Turns in Repentance
Girl/Female
Australian, Christian, Danish, French, German, Latin
Little Laurel; The Laurel Tree; Sweet Bay Tree; Symbolic of Honor and Victory
Girl/Female
Tamil
Girl/Female
Indian
Scent, Perfume
Boy/Male
Hindu, Indian, Tamil
Lord Shiva; Lord Murugan
Boy/Male
Arabic, Muslim
Tall; Towering; Lofty
Boy/Male
American, British, English
Healer; Variant of Names Like Jason and Jacob
INFINITE DIMENSIONAL-VECTOR-FUNCTION
INFINITE DIMENSIONAL-VECTOR-FUNCTION
INFINITE DIMENSIONAL-VECTOR-FUNCTION
INFINITE DIMENSIONAL-VECTOR-FUNCTION
INFINITE DIMENSIONAL-VECTOR-FUNCTION
a.
Without limit in power, capacity, knowledge, or excellence; boundless; immeasurably or inconceivably great; perfect; as, the infinite wisdom and goodness of God; -- opposed to finite.
n.
Endless or indefinite number; great multitude; as an infinity of beauties.
a.
Having dimensions.
a.
Unlimited or boundless, in time or space; as, infinite duration or distance.
n.
A literal factor, as numbered in characterizing a term. The term dimensions forms with the cardinal numbers a phrase equivalent to degree with the ordinal; thus, a2b2c is a term of five dimensions, or of the fifth degree.
n.
That which is infinite; boundless space or duration; infinity; boundlessness.
n.
An infinite quantity or magnitude.
n.
The Infinite Being; God; the Almighty.
a.
Infinite; perpetual, as a canon whose end leads back to the beginning. See Infinite, a., 5.
a.
Pertaining to dimension.
a.
Boundless; infinite.
n.
The degree of manifoldness of a quantity; as, time is quantity having one dimension; volume has three dimensions, relative to extension.
n.
The ratio of one vector to another in length, no regard being had to the direction of the two vectors; -- so called because considered as a stretching factor in changing one vector into another. See Versor.
n.
Infinite extent; unlimited space; immensity; infinity.
a.
Having certain or distinct; determinate in extent or greatness; limited; fixed; as, definite dimensions; a definite measure; a definite period or interval.
n.
An infinity; an incalculable or very great number.
pl.
of Infinity
a.
Having no determined or certain limits; large and unmeasured, though not infinite; unlimited; as indefinite space; the indefinite extension of a straight line.
n.
Same as Radius vector.
a.
Having a limit; limited in quantity, degree, or capacity; bounded; -- opposed to infinite; as, finite number; finite existence; a finite being; a finite mind; finite duration.