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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
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
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)
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
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
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)
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)
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
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
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
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)
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
Element of a basis for a function space
basis functions. In finite-dimensional vector spaces, this representation is purely algebraic and involves only finitely many basis functions, whereas
Basis_function
Smooth approximation of one-hot arg max
the original, probably highly-dimensional, input to vectors in a K-dimensional space ℝK. The standard softmax function is often used in the final layer
Softmax_function
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
Concept in linear algebra
vectors, row vectors, and matrices; hence, they are useful in calculations. The idea of a coordinate vector can also be used for infinite-dimensional
Coordinate_vector
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
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
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
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
Normed vector space that is complete
finite-dimensional. In particular, no infinite–dimensional normed space can be locally compact or have the Heine–Borel property. If x0 is a vector and s≠0
Banach_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
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
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
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
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
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
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
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
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)
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
Mathematical function, in linear algebra
linear operator is a particular kind of function between vector spaces, which respects the basic operations of vector addition and scalar multiplication.
Linear_map
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
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 (nowhere-differentiable function)
Weierstrass_function_(nowhere-differentiable_function)
Degree of differentiability of a function or map
The function f is said to be infinitely differentiable, smooth, or of class C∞, if it is of class Ck for every non-negative integer k. The function f is
Smoothness
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
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
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
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 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 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
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)
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
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
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)
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
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
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
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
Coordinate change in linear algebra
mainly with finite-dimensional vector spaces. However, many of the presented principles are also valid for infinite-dimensional vector spaces. Let Bold=(v1
Change_of_basis
Equations of motion for viscous fluids
concerns whether they have smooth (meaning infinitely differentiable) or bounded solutions in three-dimensional Euclidean space, as opposed to a breakdown
Navier–Stokes_equations
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
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
Geometric model of the physical space
scalar and vector, and they were first defined in the three dimensional sense within his geometric framework for quaternions. Three dimensional space could
Three-dimensional_space
Operation in 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
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
Artificial neural network node function
where 𝐜 is the vector representing the function center and a and σ are parameters affecting the spread of the radius. Periodic functions can serve as activation
Activation_function
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
Algebraic structure used in analysis
algebra 𝔰𝔩(2,ℂ). The Lie algebra of vector fields on a smooth manifold of positive dimension is an infinite-dimensional Lie algebra over ℝ. The Kac–Moody
Lie_algebra
Mathematical operation on vector spaces
product of X and Y. If V and W are vector spaces of finite dimension, then V⊗W is finite-dimensional, and its dimension is the product of the dimensions
Tensor_product
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
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
Mathematical model in quantum mechanics
In general for an n-dimensional box, the solutions are ψ=∏iψni(xi,t,Li) The n-dimensional momentum wave functions may likewise be represented
Particle_in_a_box
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
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
Instantaneous rate of change (mathematics)
variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector. A function of a real variable f(x) is differentiable
Derivative
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
Algorithm for supervised learning of binary classifiers
classifiers. A binary classifier is a function that can decide whether or not an input, represented by a vector of numbers, belongs to some specific class
Perceptron
Four-dimensional number system
Quaternions can be used to represent vectors in three-dimensional space, which provides a definition of the quotient of two vectors. Quaternions were first described
Quaternion
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
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
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)
Mathematics concept
{H}}} (either finite or infinite dimensional), its complex conjugate H ¯ {\displaystyle {\overline {\mathcal {H}}}} is the same vector space as its continuous
Complex conjugate of a vector space
Complex_conjugate_of_a_vector_space
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
Point to which functions converge in analysis
example, the function concerned are finite-dimension vector-valued function. In this case, the limit theorem for vector-valued function states that if
Limit_of_a_function
Fourier transform of the probability density function
univariate distributions, characteristic functions can be defined for multivariate probability distributions of vector- or matrix-valued random variables,
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Area of functional analysis and convex analysis
is a subset of a real vector space V, and the main thrust of the theory is to treat the cases where V is an infinite-dimensional (locally convex Hausdorff)
Choquet_theory
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
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-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
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
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
topological vector spaces), then it makes sense to ask whether all linear maps are continuous. It turns out that for maps defined on infinite-dimensional topological
Discontinuous_linear_map
Problem in optimization theory
in Infinite-Dimensional Spaces. Wiley. ISBN 0-471-91250-6. OCLC 15053031. Bonnans, J. Frédéric; Shapiro, Alexander (2000). "5.4, 7.4.4 Semi-infinite programming"
Semi-infinite_programming
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
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
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
In mathematics, invariant of square matrices
parallelotope defined by n vectors in an n-dimensional Euclidean vector space is at most the product of the lengths of the vectors, and the equality holds
Determinant
Certain vector fields are the sum of an irrotational and a solenoidal vector field
Oliver (2023). "Helmholtz decomposition and potential functions for n-dimensional analytic vector fields". Journal of Mathematical Analysis and Applications
Helmholtz_decomposition
Matrix of inner products of vectors
is the Gram matrix of some vectors b(1),…,b(n). Such vectors are called a vector realization of M. The infinite-dimensional analog of this statement is
Gram_matrix
Mathematical concept
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
Linear algebra concept
span of the dual set. The dual of an infinite-dimensional space has greater dimension (this being a greater infinite cardinality) than the original space
Dual_basis
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
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
Vector space with generalized dot product
space is a real or complex vector space endowed with an operation called an inner product. The inner product of two vectors in the space is a scalar, often
Inner_product_space
Mathematical form
∀w∈W∗, where V* and W* denote the dual spaces of V and W. For infinite-dimensional vector spaces, one also has the: Tensor product of Hilbert spaces Topological
Product_(mathematics)
On the dimension of vector space duals
theorem makes a fundamental statement about the dimension of the dual spaces of infinite-dimensional vector spaces; in particular, it shows that the algebraic
Erdős–Kaplansky_theorem
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
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
German mathematician (born 1958)
this site is supposed to be endowed with an infinite-dimensional cohomology theory such that the L-function of a motive M is given by L ( M , s ) = ∏ i
Christopher_Deninger
travel, tourism, insurance
INFINITE DIMENSIONAL-VECTOR-FUNCTION
INFINITE DIMENSIONAL-VECTOR-FUNCTION
INFINITE DIMENSIONAL-VECTOR-FUNCTION
INFINITE DIMENSIONAL-VECTOR-FUNCTION
INFINITE DIMENSIONAL-VECTOR-FUNCTION
INFINITE DIMENSIONAL-VECTOR-FUNCTION
INFINITE DIMENSIONAL-VECTOR-FUNCTION
INFINITE DIMENSIONAL-VECTOR-FUNCTION
INFINITE DIMENSIONAL-VECTOR-FUNCTION
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