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Vector space with a notion of nearness
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures
Topological_vector_space
Structure in functional analysis
analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get
Complete topological vector space
Complete_topological_vector_space
Space with topology generated by convex sets
convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can
Locally convex topological vector space
Locally_convex_topological_vector_space
Topological vector space whose topology can be defined by a metric
pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit
Metrizable topological vector space
Metrizable_topological_vector_space
Vector space on which a distance is defined
put it more abstractly every seminormed vector space is a topological vector space and thus carries a topological structure which is induced by the semi-norm
Normed_vector_space
In mathematics, vector space of linear forms
be called the algebraic dual space. When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear
Dual_space
Locally convex topological vector space that is also a complete metric space
Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that
Fréchet_space
Algebraic structure in linear algebra
the case of topological vector spaces, which include function spaces, inner product spaces, normed spaces, Hilbert spaces and Banach spaces. In this article
Vector_space
Generalization of boundedness
Kolmogorov in 1935. Suppose X {\displaystyle X} is a topological vector space (TVS) over a topological field K . {\displaystyle \mathbb {K} .} A subset B
Bounded set (topological vector space)
Bounded_set_(topological_vector_space)
Length in a vector space
cases, this topological vector space is not locally convex, and has no continuous non-zero linear forms. Thus the topological dual space contains only
Norm_(mathematics)
Mathematical function
topological vector space is locally convex if and only if its topology is induced by a family of seminorms. Let X {\displaystyle X} be a vector space
Seminorm
Algebraic structure formed from a collection of algebraic structures
abelian groups have additional structure (for example, are vector spaces, modules, or topological abelian groups), then the direct sum also has that structure
Direct_sum
analysis and order theory, an ordered topological vector space, also called an ordered TVS, is a topological vector space (TVS) X that has a partial order
Ordered topological vector space
Ordered_topological_vector_space
Normed vector space that is complete
if it is complete as a topological vector space. If ( X , τ ) {\displaystyle (X,\tau )} is a metrizable topological vector space (such as any norm induced
Banach_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
Mathematical set with some added structure
and topological structures underlie the linear topological space (in other words, topological vector space) structure. A linear topological space is both
Space_(mathematics)
Function spaces generalizing finite-dimensional p norm spaces
Riesz (Riesz 1910). Lp spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role
Lp_space
Area of mathematics using condensed sets
commutative algebra over topological rings." The fundamental idea in the development of the theory is given by replacing topological spaces by condensed sets
Condensed_mathematics
Generalization of compactness
is a compact complete set that is not closed. Any topological vector space is an abelian topological group under addition, so the above conditions apply
Totally_bounded_space
Generalization of finite-dimensional Euclidean spaces different from Hilbert spaces
mathematics, nuclear spaces are topological vector spaces that can be viewed as a generalization of finite-dimensional Euclidean spaces and share many of
Nuclear_space
functional analysis and related areas of mathematics, Schwartz spaces are topological vector spaces (TVS) whose neighborhoods of the origin have a property similar
Schwartz topological vector space
Schwartz_topological_vector_space
Concept in functional analysis
subspace of a topological vector space X , {\displaystyle X,} is a vector subspace M {\displaystyle M} for which there exists some other vector subspace N
Complemented_subspace
Topological vector space
{C}}} is either the category of topological spaces or some subcategory of the category of topological vector spaces (TVSs); If all objects in the category
LF-space
Broad concept generalizing scalars in mathematics and physics
bilinear form Topological vector space, a blend of topological structure with the algebraic concept of a vector space A vector field is a vector-valued function
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Mathematical space with a notion of closeness
such as topological groups, topological rings, topological fields and topological vector spaces over the latter. Local fields are topological fields important
Topological_space
Mathematical concept
certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used
Weak_topology
Vector space with a partial order
ordered vector space or partially ordered vector space is a real vector space equipped with a partial order that is compatible with the vector space operations
Ordered_vector_space
Mathematical space with a notion of distance
claimed to be homeomorphic to the topological quotient. Goreham, Anthony. Sequential convergence in Topological Spaces Archived 2011-06-04 at the Wayback
Metric_space
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
Group that is a topological space with continuous group operations
example, in physics. In functional analysis, every topological vector space is an additive topological group with the additional property that scalar multiplication
Topological_group
also refer to a property of topological vector spaces, or of functions from a topological space into a topological vector space (TVS). A subset B ⊆ X {\displaystyle
Local_boundedness
Topological space that is homeomorphic to a metric space
mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space ( X , τ ) {\displaystyle (X
Metrizable_space
Locally convex topological vector space
mathematics known as functional analysis, a reflexive space is a locally convex topological vector space for which the canonical evaluation map from X {\displaystyle
Reflexive_space
Linear map from a vector space to its field of scalars
a linear map 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
Linear_form
Set of functions between two fixed sets
linear structure. Specifically, some are topological vector spaces, some are Banach spaces, some are Hilbert spaces, etc. This allows mathematicians to apply
Function_space
In mathematics, a completely metrizable space (metrically topologically complete space) is a topological space (X, T) for which there exists at least one
Completely_metrizable_space
Theorem on extension of bounded linear functionals
extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space. The theorem also shows that there are sufficient
Hahn–Banach_theorem
Concepts in convex analysis
if C {\displaystyle C} is neither convex nor a cone. If X is a topological vector space over the real or complex numbers, then the dual cone of a subset
Dual_cone_and_polar_cone
"Small" subset of a topological space
} is not a meagre topological space). A countable Hausdorff space without isolated points is meagre, whereas any topological space that contains an isolated
Meagre_set
a topological vector lattice that is also a normed space whose unit ball is a solid set. Normed lattices are important in the theory of topological vector
Normed_vector_lattice
Space where bounded operators are continuous
bounded subset Bounded set (topological vector space) – Generalization of boundedness Locally convex topological vector space – Space with topology generated
Bornological_space
Subject in mathematics
measure theory in topological vector spaces refers to the extension of measure theory to topological vector spaces. Such spaces are often infinite-dimensional
Measure theory in topological vector spaces
Measure_theory_in_topological_vector_spaces
Type of function in linear algebra
Metrizable topological vector space – Topological vector space whose topology can be defined by a metric Norm (mathematics) – Length in a vector space Seminorm –
Quasinorm
convex vector lattice (LCVL) is a topological vector lattice that is also a locally convex space. LCVLs are important in the theory of topological vector lattices
Locally_convex_vector_lattice
Theorem stating that pointwise boundedness implies uniform boundedness
metrizable topological vector space X {\displaystyle X} (such as a Fréchet space or an F-space) into a Hausdorff topological vector space Y . {\displaystyle
Uniform_boundedness_principle
Vector space of infinite sequences
subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space. The most important
Sequence_space
Type of topological vector space
mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood
Barrelled_space
Topological vector space with a complete translation-invariant metric
(functional analysis) LB-space LF-space – Topological vector space Metrizable topological vector space – Topological vector space whose topology can be defined
F-space
in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) X {\displaystyle X} that has a partial
Topological_vector_lattice
Space formed by the ''n''-tuples of real numbers
multiplication, it is a real vector space. The coordinates over any basis of the elements of a real vector space form a real coordinate space of the same dimension
Real_coordinate_space
Mathematical function, in linear algebra
mapping) is a particular kind of function between vector spaces, which respects the basic operations of vector addition and scalar multiplication. A standard
Linear_map
Tensor product constructions for topological vector spaces
different ways to construct a topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved
Topological_tensor_product
Infinite sum
locally convex spaces. Berlin, New York: Springer-Verlag. ISBN 0-387-05644-0. OCLC 539541. Robertson, A. P. (1973). Topological vector spaces. Cambridge England:
Series_(mathematics)
Metric geometry
Fréchet space: a locally convex topological vector space whose topology can be induced by a complete translation-invariant metric. The space Q p {\displaystyle
Complete_metric_space
General concept and operation in mathematics
In the realm of topological vector spaces, a similar construction exists, replacing the dual by the topological dual vector space. There are several
Duality_(mathematics)
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
Topological vector space in which every closed and bounded subset is complete
In functional analysis, a topological vector space (TVS) is said to be quasi-complete or boundedly complete if every closed and bounded subset is complete
Quasi-complete_space
Collection of mathematical objects of finite size
the topological vector space is induced by a metric which is homogeneous, as in the case of a metric induced by the norm of normed vector spaces, then
Bounded_set
mathematics known as functional analysis, a semi-reflexive space is a locally convex topological vector space (TVS) X such that the canonical evaluation map from
Semi-reflexive_space
On when a space equals the closed convex hull of its extreme points
convex topological vector spaces (TVSs). Krein–Milman theorem—A compact convex subset of a Hausdorff locally convex topological vector space is equal
Krein–Milman_theorem
In geometry, set whose intersection with every line is a single line segment
transformations. Further, it implies that a convex set in a real or complex topological vector space is path-connected (and therefore also connected). A set C is strictly
Convex_set
Barrelled space where closed and bounded subsets are compact
Montel space is a barrelled topological vector space in which every closed and bounded subset is compact. A topological vector space (TVS) has the Heine–Borel
Montel_space
In functional analysis, a topological vector space (TVS) X {\displaystyle X} is called ultrabornological if every bounded linear operator from X {\displaystyle
Ultrabornological_space
Topological space with a notion of uniform properties
continuous function. Every topological group G {\displaystyle G} (in particular, every topological vector space) becomes a uniform space if we define a subset
Uniform_space
Construction in category theory
carried out if the A i {\displaystyle A_{i}} 's are sets, semigroups, topological spaces, rings, modules (over a fixed ring), algebras (over a fixed ring)
Inverse_limit
functional analysis, a discipline within mathematics, a locally convex topological vector space (TVS) is said to be infrabarrelled (also spelled infrabarreled)
Infrabarrelled_space
Area of mathematics
theorem—If X {\displaystyle X} is a topological vector space and Y {\displaystyle Y} is a compact Hausdorff topological vector space, then the graph of a linear
Functional_analysis
Sequence of points that get progressively closer to each other
Cauchy sequences. There is also a concept of Cauchy sequence for a topological vector space X {\displaystyle X} : Pick a local base B {\displaystyle B} for
Cauchy_sequence
Mathematical function
space of X ⊗ ^ ϵ Y {\displaystyle X{\widehat {\otimes }}_{\epsilon }Y} , the injective tensor product of the locally convex topological vector spaces
Integral_linear_operator
Vector space with generalized dot product
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
Inner_product_space
Subset of Euclidean space is compact if and only if it is closed and bounded
Banach spaces have the Heine–Borel property (as topological vector spaces). But some infinite-dimensional Fréchet spaces do have, for instance, the space C
Heine–Borel_theorem
Theorem of convex functions
or the topological vector space are needed, see Example (1.3) on p. 53 in Perlman, Michael D. (1974). "Jensen's Inequality for a Convex Vector-Valued
Jensen's_inequality
French mathematician (1928–2014)
His key contributions include topological tensor products of topological vector spaces, the theory of nuclear spaces as foundational for Schwartz distributions
Alexander_Grothendieck
Topological space that is connected
Connectedness is one of the principal topological properties that distinguish topological spaces. A subset of a topological space X {\displaystyle X} is a connected
Connected_space
this article discusses topologies on such spaces in the more general setting of topological vector spaces (TVSs). Throughout, the following is assumed:
Topologies on spaces of linear maps
Topologies_on_spaces_of_linear_maps
Type of mathematical space
corresponding topological property is used to define compactness: a topological space is compact if every open cover has a finite subcover. In metric spaces this
Compact_space
Topological space characterized by sequences
space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that
Sequential_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
hypercyclic operator on a topological vector space X is a continuous linear operator T: X → X such that there is a vector x ∈ X for which the sequence
Hypercyclic_operator
Concept in functional analysis
a topological homomorphism or simply homomorphism (if no confusion will arise) is the analog of homomorphisms for the category of topological vector spaces
Topological_homomorphism
Function between topological vector spaces
a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear operator if and only if
Continuous_linear_operator
over metrizable spaces. Every complete topological vector space is quasi-complete and every quasi-complete topological vector space is sequentially complete
Sequentially_complete
Kind of linear transformation
linear transformation L : X → Y {\displaystyle L:X\to Y} between topological vector spaces (TVSs) X {\displaystyle X} and Y {\displaystyle Y} that maps bounded
Bounded_operator
X {\displaystyle X} is a topological vector space then the set of all bounded subsets of X {\displaystyle X} from a vector bornology on X {\displaystyle
Vector_bornology
In functional analysis, a topological vector space (TVS) is said to be countably quasi-barrelled if every strongly bounded countable union of equicontinuous
Countably quasi-barrelled space
Countably_quasi-barrelled_space
absorbing then the two auxiliary normed spaces are canonically isomorphic (as topological vector spaces and as normed spaces). Throughout this article, X {\displaystyle
Auxiliary_normed_space
Set of vectors used to define coordinates
In mathematics, a set B of elements of a vector space V is called a basis (pl.: bases) if every element of V can be written in a unique way as a finite
Basis_(linear_algebra)
Construct in functional analysis
functional analysis because every neighborhood of the origin in every topological vector space (TVS) contains a balanced neighborhood of the origin and every
Balanced_set
Type of vector space in math
plane and three-dimensional space to spaces of any finite or infinite dimension. A Hilbert space is an abstract vector space, and it has the additional
Hilbert_space
Certain topology in mathematics
union of (possibly infinitely many) such open intervals and rays. A topological space X is called orderable or linearly orderable if there exists a total
Order_topology
Type of topological space in mathematics
a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More
Locally_compact_space
Extension of the Brouwer fixed-point theorem
extension of the Brouwer fixed-point theorem to locally convex topological vector spaces, which may be of infinite dimension. It asserts that if K {\displaystyle
Schauder_fixed-point_theorem
In functional analysis, a topological vector space (TVS) is said to be countably barrelled if every weakly bounded countable union of equicontinuous subsets
Countably_barrelled_space
Branch of mathematics
x_{n})\mapsto a_{1}x_{1}+\cdots +a_{n}x_{n},} and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas
Linear_algebra
Continuous dual space endowed with the topology of uniform convergence on bounded sets
mathematics, the strong dual space of a topological vector space (TVS) X {\displaystyle X} is the continuous dual space X ′ {\displaystyle X^{\prime }}
Strong_dual_space
Duality for locally compact abelian groups
finite-dimensional vector space over the reals or a p-adic field. The Pontryagin dual of a locally compact abelian group is the locally compact abelian topological group
Pontryagin_duality
Relation among continuous functions
topological space and Y is a uniform space given by family of entourages U {\displaystyle {\mathcal {U}}} . Topological vector spaces, metric spaces and
Equicontinuity
List of concrete topologies and topological spaces
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This
List_of_topologies
Characterization of normable spaces
and sufficient condition for a topological vector space to be normable; that is, for the existence of a norm on the space that generates the given topology
Kolmogorov's normability criterion
Kolmogorov's_normability_criterion
topology, named after George Mackey, is the finest topology for a topological vector space which still preserves the continuous dual. In other words the Mackey
Mackey_topology
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