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This page lists some examples of vector spaces. See vector space for the definitions of terms used on this page. See also: dimension, basis. Notation.
Examples_of_vector_spaces
Algebraic structure in linear algebra
operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. Real vector spaces and complex vector spaces are
Vector_space
Vector space on which a distance is defined
of normed spaces and Banach spaces is a fundamental part of functional analysis, a major subfield of mathematics. A normed vector space is a vector space
Normed_vector_space
Topics referred to by the same term
vector, a vector where all components are zero 0 vector space; see Examples of vector spaces 0-velocity surface, or Zero-velocity surface V0 (disambiguation)
0V
Broad concept generalizing scalars in mathematics and physics
coordinate vector space. Many vector spaces are considered in mathematics, such as extension fields, polynomial rings, algebras and function spaces. The term
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Vector space with a notion of nearness
spaces and Sobolev spaces. Many topological vector spaces are spaces of functions, or linear operators acting on topological vector spaces, and the topology
Topological_vector_space
Mathematical parametrization of vector spaces by another space
mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X {\displaystyle
Vector_bundle
Algebraic structure formed from a collection of algebraic structures
(for example, are vector spaces, modules, or topological abelian groups), then the direct sum usually maintains that structure (as an example of the opposite
Direct_sum
Mathematical property
one-dimensional vector spaces are the simple ones. So it is a basic result of linear algebra that any finite-dimensional vector space is the direct sum of simple
Semi-simplicity
of topics related to linear algebra, the branch of mathematics concerning linear equations and linear maps and their representations in vector spaces
Outline_of_linear_algebra
Assignment of a vector to each point in a subset of Euclidean space
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space R n {\displaystyle
Vector_field
Vector space with generalized dot product
Euclidean vector spaces, in which the inner product is the dot product or scalar product of Cartesian coordinates. Inner product spaces of infinite dimensions
Inner_product_space
Fundamental space of geometry
spaces through axiomatic theory. Another definition of Euclidean spaces by means of vector spaces and linear algebra has been shown to be equivalent to
Euclidean_space
Space with topology generated by convex sets
related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that
Locally convex topological vector space
Locally_convex_topological_vector_space
Algebraic structure with only one element
Nildimensional space Triviality (mathematics) Examples of vector spaces Field with one element Empty semigroup Zero element List of zero terms David
Zero_object_(algebra)
Examples of generating functions List of space groups Examples of Markov chains Examples of vector spaces Fano plane Frieze group Gray graph Hall–Janko graph
List_of_mathematical_examples
Algebraic structure decomposed into a direct sum
the vector space into a direct sum of vector subspaces, generally indexed by the integers. For "pure" vector spaces, the concept has been introduced in
Graded_vector_space
Normed vector space that is complete
Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. A Banach space is a
Banach_space
Mathematical function, in linear algebra
kind of function between vector spaces, which respects the basic operations of vector addition and scalar multiplication. A standard example of a linear
Linear_map
Euclidean space without distance and angles
sources define affine spaces in terms of the well developed vector space theory. An affine space is a set A together with a vector space A → {\displaystyle
Affine_space
Number of vectors in any basis of the vector space
In mathematics, the dimension of a vector space V is the cardinality (i.e., the number of vectors) of a basis of V over its base field. It is sometimes
Dimension_(vector_space)
Type of vector space in math
success of Hilbert space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean vector spaces, examples of Hilbert
Hilbert_space
vector space V {\displaystyle \;V} is a vector space together with an additional structure map τ {\displaystyle \tau } symbolizing interchanging of two
Braided_vector_space
Set of vectors used to define coordinates
vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. Basis vectors find applications in the study of
Basis_(linear_algebra)
Geometric object that has length and direction
Euclidean vectors can be added and scaled to form a vector space. A vector quantity is a vector-valued physical quantity, including units of measurement
Euclidean_vector
Mathematical operation on vector spaces
{\displaystyle V\otimes W} of two vector spaces V {\displaystyle V} and W {\displaystyle W} (over the same field) is a vector space to which is associated
Tensor_product
Area of mathematics using condensed sets
"liquid vector space in nLab". ncatlab.org. Retrieved 2023-11-07. Scholze, Peter. "Lectures on Analytic Geometry: Lecture III: Condensed ℝ-vector spaces" (PDF)
Condensed_mathematics
Elements of a field, e.g. real numbers, in the context of linear algebra
is called an inner product space. A scalar may also have other roles in terms of vector components, in normed vector spaces, in modules, and in transformations
Scalar_(mathematics)
Physical quantity that is a vector
the natural sciences, a vector quantity (also known as a vector physical quantity, physical vector, or simply vector) is a vector-valued physical quantity
Vector_quantity
In mathematics, vector space of linear forms
finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe
Dual_space
Vectors whose components are all 0 except one that is 1
corresponding Cartesian coordinate system. Canonical units Examples of vector spaces § Generalized coordinate space Roman (2008), p. 47, ch. 1. Axler (2015), pp. 39–40
Standard_basis
Model for representing text documents
Vector space model (VSM) or term vector model is an algebraic model for representing text documents (or more generally, items) as vectors such that the
Vector_space_model
Length in a vector space
{\langle x,x\rangle }}.} Other examples of infinite-dimensional normed vector spaces can be found in the Banach space article. Generally, these norms
Norm_(mathematics)
Locally convex topological vector space that is also a complete metric space
(normed vector spaces that are complete with respect to the metric induced by the norm). All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely
Fréchet_space
Function of two vectors linear in each argument
elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. A bilinear
Bilinear_map
Vectors whose linear combinations are nonzero
subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space. A sequence of vectors from a vector space
Linear_independence
Banach space with a compatible structure of a lattice
X)(f(x)\leq g(x)){\text{.}}} Examples of non-lattice Banach spaces are now known; James' space is one such. The continuous dual space of a Banach lattice is equal
Banach_lattice
Mathematical space with two coordinates
two-dimensional spaces are often called planes (especially the Euclidean plane), or, more generally, surfaces. These include analogs to physical spaces, like flat
Two-dimensional_space
Calculus of vector-valued functions
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional
Vector_calculus
Concept in linear algebra
idea 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
Topological space in group theory
group of X. Riemannian symmetric spaces are an important class of homogeneous spaces, and include many of the examples listed below. Concrete examples include:
Homogeneous_space
Completion of the usual space with "points at infinity"
affine space with a distinguished point O may be identified with its associated vector space (see Affine space § Vector spaces as affine spaces), the preceding
Projective_space
Function acting on function spaces
examples of infinite-dimensional vector spaces). The space of sequences of real numbers, or more generally sequences of vectors in any vector space,
Operator_(mathematics)
dense orbit in V. The term prehomogeneous vector space was introduced by Mikio Sato in 1970. These spaces have many applications in geometry, number
Prehomogeneous_vector_space
Choice of reference for distinguishing an object and its mirror image
The orientation of a real vector space or simply orientation of a vector space is the arbitrary choice of which ordered bases are "positively" oriented
Orientation_(vector_space)
Mathematical concept
mathematics, a symplectic vector space is a vector space V {\displaystyle V} over a field F {\displaystyle F} (for example the real numbers R {\displaystyle
Symplectic_vector_space
Set of functions between two fixed sets
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
Function_space
Vector spaces associated to a matrix
possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding matrix transformation. Let
Row_and_column_spaces
Category whose objects are R-modules and whose morphisms are module homomorphisms
important invariant of the category of modules.) Category of rings Derived category Module spectrum Category of graded vector spaces Category of representations
Category_of_modules
Whose values lie in an infinite-dimensional vector space
infinite-dimensional vector function is a function whose values lie in an infinite-dimensional topological vector space, such as a Hilbert space or a Banach space. Such
Infinite-dimensional vector function
Infinite-dimensional_vector_function
Vector space equipped with a bilinear product
is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication
Algebra_over_a_field
Algebraic object with geometric applications
many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some
Tensor
Operation in abstract algebra
familiar examples of this construction occur when considering vector spaces (modules over a field) and abelian groups (modules over the ring Z of integers)
Direct_sum_of_modules
Assignment of vector fields to manifolds
at any point is equal to the tangent vector attached to that point by the vector field. All the tangent spaces of a manifold may be "glued together" to
Tangent_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
Mathematical space with a notion of distance
to another. Metric spaces appear in many different branches of mathematics. For example, Riemannian manifolds, normed vector spaces, and graphs may be
Metric_space
functional analysis, spaces of linear maps between two vector spaces can be endowed with a variety of topologies. Studying space of linear maps and these
Topologies on spaces of linear maps
Topologies_on_spaces_of_linear_maps
Generalization of boundedness
related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be inflated
Bounded set (topological vector space)
Bounded_set_(topological_vector_space)
Matrix consisting of a single row or column
Similarly, a row vector is a 1 × n {\displaystyle 1\times n} matrix, consisting of a single row of n {\displaystyle n} entries. For example, x {\displaystyle
Row_and_column_vectors
Generalization of finite-dimensional Euclidean spaces different from Hilbert spaces
nuclear spaces are topological vector spaces that can be viewed as a generalization of finite-dimensional Euclidean spaces and share many of their desirable
Nuclear_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
In mathematics, vector subspace
linear algebra, a linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply
Linear_subspace
Mathematical set with some added structure
subset of the parent space which retains the same mathematical structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear
Space_(mathematics)
Vector behavior under coordinate changes
the dual vector space, and represents a linear map from vectors to scalars. The dot product operator involving vectors is a good example of a covector
Covariance and contravariance of vectors
Covariance_and_contravariance_of_vectors
Property of a mathematical space
High-dimensional spaces frequently occur in mathematics and the sciences. They may be Euclidean spaces or more general parameter spaces or configuration spaces such
Dimension
Mathematical measure of how much a curve or surface deviates from flatness
makes senses in metric spaces, and this gives rise to CAT(k) spaces. Curvature form for the appropriate notion of curvature for vector bundles and principal
Curvature
Multivariate derivative (mathematics)
In vector calculus, the gradient of a scalar-valued differentiable function f {\displaystyle f} of several variables is the vector field (or vector-valued
Gradient
Physical spaces representing position and momentum, Fourier-transform duals
vector spaces, usually three-dimensional but in general of any finite dimension. Position space (also real space or coordinate space) is the set of all
Position_and_momentum_spaces
Geometric space with six dimensions
properties of all Euclidean spaces, so it is linear, has a metric and a full set of vector operations. In particular the dot product between two 6-vectors is
Six-dimensional_space
Linear approximation of smooth maps on tangent spaces
geometry, pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces. Suppose that φ : M → N {\displaystyle \varphi
Pushforward_(differential)
Property determining comparison and ordering
number and zero. In vector spaces, the Euclidean norm is a measure of magnitude used to define a distance between two points in space. In physics, magnitude
Magnitude_(mathematics)
Vector space consisting of affine subsets
linear algebra, the quotient of a vector space V {\displaystyle V} by a subspace U {\displaystyle U} is a vector space obtained by "collapsing" U {\displaystyle
Quotient space (linear algebra)
Quotient_space_(linear_algebra)
Generalization of compactness
statement 6(a) was the first reformulation of total boundedness for topological vector spaces; it dates to a 1935 paper of John von Neumann. This definition has
Totally_bounded_space
Concept in mathematics
extensions of basic integral transforms into more abstract spaces, vector-valued functions, and operator spaces. Examples of such extensions include vector-valued
Bochner_integral
Area of mathematics
branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, inner
Functional_analysis
Geometric model of the physical space
origin' of the vector space. Euclidean spaces are sometimes called Euclidean affine spaces for distinguishing them from Euclidean vector spaces. This is
Three-dimensional_space
quasibarrelled spaces are topological vector spaces (TVS) for which every bornivorous barrelled set in the space is a neighbourhood of the origin. Quasibarrelled
Infrabarrelled_space
Partially ordered vector space, ordered as a lattice
Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice. Riesz spaces are
Riesz_space
Property of geometry, also used to generalize the notion of "distance" in metric spaces
is norm in any inner product space, a generalization of Euclidean vector spaces including infinite-dimensional examples. The triangle inequality follows
Triangle_inequality
Non-tensorial representation of the spin group
elements of a complex vector space that can be associated with Euclidean space. Spinors can be thought of as companion geometric objects to Euclidean space that
Spinor
Topic in mathematics
mathematics, the complexification of a vector space V over the field of real numbers (a "real vector space") yields a vector space VC over the complex number
Complexification
Topological vector space whose topology can be defined by a metric
functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced
Metrizable topological vector space
Metrizable_topological_vector_space
Method in natural language processing
meaning of the word in such a way that the words that are closer in the vector space are expected to be similar in meaning. Word embeddings can be obtained
Word_embedding
Topological space that is homeomorphic to a metric space
space to be metrizable. Metrizable spaces inherit all topological properties from metric spaces. For example, they are Hausdorff paracompact spaces (and
Metrizable_space
Vector on which a quadratic form is zero
been used in quadratic spaces, and anisotropic space for a quadratic space without null vectors. A pseudo-Euclidean vector space may be decomposed (non-uniquely)
Null_vector
Vector in relativity
special relativity, a four-vector (or 4-vector, sometimes Lorentz vector) is an element of a four-dimensional vector space object with four components
Four-vector
Vector satisfying some of the criteria of an eigenvector
be an n {\displaystyle n} -dimensional vector space and let A {\displaystyle A} be the matrix representation of a linear map from V {\displaystyle V} to
Generalized_eigenvector
Linear map from a vector space to its field of scalars
from a vector space to its field of scalars (often, the real numbers or the complex numbers). If V is a vector space over a field k, the set of all linear
Linear_form
continuous. A general version of the closed graph theorem holds for ultrabornological spaces. Ultrabornological spaces were introduced by Alexander Grothendieck
Ultrabornological_space
In mathematics, the category of topological vector spaces is the category whose objects are topological vector spaces and whose morphisms are continuous
Category of topological vector spaces
Category_of_topological_vector_spaces
Structure in functional analysis
topological vector space to possess. The notions of completeness for normed spaces and metrizable TVSs, which are commonly defined in terms of completeness of a
Complete topological vector space
Complete_topological_vector_space
Assignment of a tensor continuously varying across a region of space
generalization of a scalar field and a vector field that assigns, respectively, a scalar or vector to each point of space. If a tensor A is defined on a vector fields
Tensor_field
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
Mathematics concept
George Mackey. Examples of locally convex spaces that are Mackey spaces include: All barrelled spaces and more generally all infrabarreled spaces Hence in particular
Mackey_space
Measurable property or characteristic
vector space associated with these vectors is often called the feature space. In order to reduce the dimensionality of the feature space, a number of
Feature_(machine_learning)
General concept and operation in mathematics
evaluation map". For finite-dimensional vector spaces this is an isomorphism, but these are not identical spaces: they are different sets. In category theory
Duality_(mathematics)
Algorithm on pulse-width modulation
Space vector modulation (SVM) is an algorithm for the control of pulse-width modulation (PWM), invented by Gerhard Pfaff, Alois Weschta, and Albert Wick
Space_vector_modulation
Mathematical space with a notion of closeness
Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, topological spaces are fundamental and are
Topological_space
Set of methods for supervised statistical learning
In machine learning, a support vector machine (SVM) or support vector network is a supervised max-margin model with associated learning algorithms that
Support_vector_machine
Space with one dimension
one-dimensional spaces but are usually referred to by more specific terms. Any field K {\displaystyle K} is a one-dimensional vector space over itself. The
One-dimensional_space
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EXAMPLES OF-VECTOR-SPACES
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EXAMPLES OF-VECTOR-SPACES
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EXAMPLES OF-VECTOR-SPACES
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