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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
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
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
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)
Algebraic structure in linear algebra
In mathematics, a vector space (also called a linear space) is a set whose elements, often called vectors, can be added together and multiplied ("scaled")
Vector_space
Broad concept generalizing scalars in mathematics and physics
on the above sorts of vectors. A vector space formed by geometric vectors is called a Euclidean vector space, and a vector space formed by tuples is called
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Vector space with a binary relation
} An Archimedean (pre)ordered vector space is a (pre)ordered vector space whose order is Archimedean. A preordered vector space X {\displaystyle X} is
Archimedean ordered vector space
Archimedean_ordered_vector_space
Partially ordered topological space
{\displaystyle x\leq y} . Ordered vector space – Vector space with a partial order Ordered topological vector space Topological vector lattice Gierz, G.; Hofmann
Partially_ordered_space
Mathematical set with an ordering
descriptions of redirect targets Ordered vector space – Vector space with a partial order Poset topology, a kind of topological space that can be defined from
Partially_ordered_set
Order whose elements are all comparable
two sets. Applied to the vector space Rn, each of these make it an ordered vector space. See also examples of partially ordered sets. A real function of
Total_order
Algebraic object with an ordered structure
space Ordered vector space – Vector space with a partial order Partially ordered ring – Ring with a compatible partial order Partially ordered space –
Ordered_field
Partially ordered space – Partially ordered topological space Riesz space – Partially ordered vector space, ordered as a lattice, also called vector lattice
Ordered_ring
Vector space on which a distance is defined
In mathematics, a normed vector space or normed space is a vector space, typically over the real or complex numbers, on which a norm is defined. A norm
Normed_vector_space
Concept in linear algebra
coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis
Coordinate_vector
Generalized alphabetical order
0]<[1,1,0]<[2,0,0]} For the lexicographical order, the same exponent vectors are ordered as [ 0 , 0 , 2 ] < [ 0 , 1 , 1 ] < [ 0 , 2 , 0 ] < [ 1 , 0 , 1 ]
Lexicographic_order
Special type of lattice
is a Boolean algebra if and only if n is square-free. A lattice-ordered vector space is a distributive lattice. Young's lattice given by the inclusion
Distributive_lattice
Physical quantity that is a vector
and a vector numerical value (unitless), often a Euclidean vector with magnitude and direction. For example, a position vector in physical space may be
Vector_quantity
continuous. If X {\displaystyle X} is a vector lattice and an ordered topological vector space that is a Fréchet space in which the positive cone is a normal
Topological_vector_lattice
Property of subsets of ordered vector spaces
theory and functional analysis, a subset A {\displaystyle A} of an ordered vector space is said to be order complete in X {\displaystyle X} if for every
Order_complete
Class of mathematical orderings
that is also maximum of the whole set. A well-ordered set as topological space is a first-countable space if and only if it has order type less than or
Well-order
Subset of incomparable elements
partially ordered set such that any two distinct elements in the subset are incomparable. The size of the largest antichain in a finite partially ordered set
Antichain
Vector space with a notion of nearness
A topological vector space is a vector space that is also a topological space with the property that the vector space operations (vector addition and scalar
Topological_vector_space
Euclidean space without distance and angles
point, the zero vector is called the origin. Adding a fixed vector to the elements of a linear subspace (vector subspace) of a vector space produces an affine
Affine_space
Well-quasi-ordering of finite trees
states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding. A finitary application
Kruskal's_tree_theorem
Fréchet lattice – Topological vector lattice Locally convex vector lattice Vector lattice – Partially ordered vector space, ordered as a latticePages displaying
Normed_vector_lattice
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
Banach space with a compatible structure of a lattice
abstract (L)-spaces. Banach space – Normed vector space that is complete Normed vector lattice Riesz space – Partially ordered vector space, ordered as a lattice
Banach_lattice
Use of coordinates for representing vectors
may be Euclidean vectors, or more generally, members of a vector space. For denoting a vector, the common typographic convention is lower case, upright
Vector_notation
Type of ordering of a set
covering relation is empty. The rational numbers as a linearly ordered set are a densely ordered set in this sense, as are the algebraic numbers, the real
Dense_order
Vector of length one
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase
Unit_vector
specifically in functional analysis, a positive linear functional on an ordered vector space ( V , ≤ ) {\displaystyle (V,\leq )} is a linear functional f {\displaystyle
Positive_linear_functional
Group with a compatible partial order
vector space Ordered vector space – Vector space with a partial order Partially ordered ring – Ring with a compatible partial order Partially ordered
Partially_ordered_group
In mathematics, vector space of linear forms
In mathematics, every vector space V {\displaystyle V} has a corresponding dual vector space (or just dual space for short) consisting of all linear forms
Dual_space
Element of an ordered vector space
An order unit is an element of an ordered vector space which can be used to bound all elements from above. In this way (as seen in the first example below)
Order_unit
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
Size of subsets in order theory
mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A. Formally
Cofinality
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)
Visual depiction of a partially ordered set
represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set ( S , ≤ ) {\displaystyle
Hasse_diagram
On chains and antichains in partial orders
combinatorics, Dilworth's theorem states that, in any finite partially ordered set, the maximum size of an antichain of incomparable elements equals the
Dilworth's_theorem
Mathematical definition
order theory and functional analysis, an ordered vector space X {\displaystyle X} is said to be regularly ordered and its order is called regular if X {\displaystyle
Regularly_ordered
(\sup S).} Vector lattice – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect targets AM-space – Concept
Abstract_L-space
Topics referred to by the same term
refer to: Positive cone of an ordered field Positive cone of an ordered vector space Positive cone of a partially ordered group This disambiguation page
Positive_cone
Order-preserving mathematical function
subspace of X . {\displaystyle X.} In functional analysis on a topological vector space X {\displaystyle X} , a (possibly non-linear) operator T : X → X ∗ {\displaystyle
Monotonic_function
Mathematical set closed under positive linear combinations
in a vector space over any ordered field, although the field of real numbers is used most often. A subset C {\displaystyle C} of a vector space is a cone
Convex_cone
Certain topology in mathematics
normal Hausdorff space. The standard topologies on R, Q, Z, and N are the order topologies. If Y is a subset of X, X a totally ordered set, then Y inherits
Order_topology
Mathematical proposition equivalent to the axiom of choice
the theorem that every vector space has a basis, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems
Zorn's_lemma
Special subset of a partially ordered set
lattice of vector subspaces of a given vector space, ordered by inclusion. Explicitly, a linear filter on a vector space X is a family B of vector subspaces
Filter_(mathematics)
Reflexive and transitive binary relation
{\displaystyle p\wedge q} , and so is q {\displaystyle q} . The partially ordered set ( X / ⇔ , ⇐ ) {\displaystyle \left(X/\Leftrightarrow ,\Leftarrow \right)}
Preorder
Geometric object that has length and direction
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
Euclidean_vector
Pair of mathematical objects
2-dimensional vectors[citation needed] (technically, this is an abuse of terminology since an ordered pair need not be an element of a vector space). The entries
Ordered_pair
Mathematical ranking of a set
generalization of totally ordered sets (rankings without ties) and are in turn generalized by (strictly) partially ordered sets and preorders. There are
Weak_ordering
Construction in order theory
on the Cartesian product of totally ordered sets Ordinal sum of partial orders Ordered vector space – Vector space with a partial order Neggers, J.; Kim
Product_order
Type of topology in mathematics
observed an equivalence between partially ordered sets and spaces that were precisely the T0 versions of the spaces that Alexandrov had introduced. P. T.
Alexandrov_topology
Property determining comparison and ordering
(vector tail) to that point (vector tip). Mathematically, a vector x in an n-dimensional Euclidean space can be defined as an ordered list of n real numbers
Magnitude_(mathematics)
Characterizes the height of any finite partially ordered set
combinatorics, Mirsky's theorem characterizes the height of any finite partially ordered set in terms of a partition of the order into a minimum number of antichains
Mirsky's_theorem
Topology of an ordered vector space
order topology of an ordered vector space ( X , ≤ ) {\displaystyle (X,\leq )} is the finest locally convex topological vector space (TVS) topology on X
Order topology (functional analysis)
Order_topology_(functional_analysis)
Algebraic object with geometric applications
of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There
Tensor
a special case of a vector optimization problem: The objective space is the finite dimensional Euclidean space partially ordered by the component-wise
Vector_optimization
Mathematical property of subsets in order theory
partially ordered set A {\displaystyle A} admits a totally ordered cofinal subset, then we can find a subset B {\displaystyle B} that is well-ordered and cofinal
Cofinal_(mathematics)
specifically in order theory and functional analysis, the order dual of an ordered vector space X {\displaystyle X} is the set Pos ( X ∗ ) − Pos ( X ∗ ) {\displaystyle
Order dual (functional analysis)
Order_dual_(functional_analysis)
Nonempty, upper-bounded, downward-closed subset
mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion
Ideal_(order_theory)
Topics referred to by the same term
principle for ordered sets Order dual (functional analysis), set of all differences of any two positive linear functionals on an ordered vector space This disambiguation
Order_dual
Branch of mathematics
partially ordered set is attributed to Garrett Birkhoff in the second edition of his influential book Lattice Theory. This section introduces ordered sets
Order_theory
Topological vector lattice
Concept in order theory Normed lattice Vector lattice – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect
Fréchet_lattice
Mathematical property of algebraic structures
999... – Alternative decimal expansion of 1 Archimedean ordered vector space – Vector space with a binary relation Construction of the real numbers "Math
Archimedean_property
Bound lattice in which every element has a complement
is bounded and relatively complemented. The lattice of subspaces of a vector space provide an example of a complemented lattice that is not, in general
Complemented_lattice
Algebraic structure used in logic
terminal object 1 ordered by inclusion, equivalently the morphisms from 1 to the subobject classifier Ω. The open sets of any topological space form a complete
Heyting_algebra
Mathematical concept
order theory and functional analysis, the order bound dual of an ordered vector space X {\displaystyle X} is the set of all linear functionals on X {\displaystyle
Order_bound_dual
Mathematical concept
In 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
Isomorphism type of ordered sets
In mathematics, especially in set theory and order theory, two ordered sets X and Y are said to have the same order type if they are order isomorphic
Order_type
considered as a vector space over R {\displaystyle \mathbb {R} } then it is an Archimedean ordered vector space. Let A be an ordered algebra with unit
Ordered_algebra
Topics referred to by the same term
an idempotent semigroup Band (order theory), a solid subset of an ordered vector space that contains its supremums Band (radio), a range of frequencies
Band
Concept in functional analysis
linear operator from an preordered vector space ( X , ≤ ) {\displaystyle (X,\leq )} into a preordered vector space ( Y , ≤ ) {\displaystyle (Y,\leq )}
Positive_linear_operator
Algebra associated to any vector space
In mathematics, the exterior algebra or Grassmann algebra of a vector space V {\displaystyle V} is an associative algebra that contains V , {\displaystyle
Exterior_algebra
Property of a relation on a set
type Ordered field Positive cone of an ordered field Ordered vector space Partially ordered Positive cone of an ordered vector space Riesz space Partially
Connected_relation
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
Ring with a compatible partial order
targets Ordered topological vector space Ordered vector space – Vector space with a partial order Partially ordered space – Partially ordered topological
Partially_ordered_ring
analysis, a discipline within mathematics, a locally convex topological vector space (TVS) is said to be infrabarrelled (also spelled infrabarreled) if every
Infrabarrelled_space
Euclidean geometry without distance and angles
vector space of the translations. In more concrete terms, this amounts to having an operation that associates to any ordered pair of points a vector and
Affine_geometry
Smallest transitive relation containing a given binary relation
relation on any set, the "less than or equal" relation on any linearly ordered set, and the relation "x was born before y" on the set of all people. Symbolically
Transitive_closure
Assignment of vector fields to manifolds
tangent space at x {\displaystyle x} are called the tangent vectors at x {\displaystyle x} . This is a generalization of the notion of a vector, based
Tangent_space
Mathematical result on order relations
{\displaystyle R} on a set X {\displaystyle X} is formally defined as a set of ordered pairs ( x , y ) {\displaystyle (x,y)} of elements of X , {\displaystyle
Szpilrajn_extension_theorem
Set whose pairs have minima and maxima
subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called
Lattice_(order)
Mathematical operation on vectors in 3D space
Euclidean vector space (named here E {\displaystyle E} ), and is denoted by the symbol × {\displaystyle \times } . Given two linearly independent vectors a and
Cross_product
Concept in order theory
order theory, the join of a subset S {\displaystyle S} of a partially ordered set P {\displaystyle P} is the supremum (least upper bound) of S , {\displaystyle
Join_and_meet
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
Structure in functional analysis
related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively
Complete topological vector space
Complete_topological_vector_space
Ideals in a Boolean algebra can be extended to prime ideals
bases of a given vector space have the same cardinality. List of Boolean algebra topics Halpern, James D. (1966), "Bases in Vector Spaces and the Axiom of
Boolean_prime_ideal_theorem
∈ S . {\displaystyle x\in S.} An ordered vector space whose order is Archimedean is said to be Archimedean ordered. If S ⊆ X {\displaystyle S\subseteq
Solid_set
Equivalence of partially ordered sets
function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can be considered
Order_isomorphism
Type of binary relation
articles for more details. Well-founded relations that are not totally ordered include: The positive integers {1, 2, 3, ...}, with the order defined by
Well-founded_relation
American automobile manufacturer
company's designer. He refused, and ordered the Vector headquarters physically locked down. He was later fired from Vector Aeromotive. The WX-3 Coupe was originally
Vector_Motors
Relationship between elements of two sets
used an indefinite inner product, and specified that a time vector is normal to a space vector when that product is zero. The indefinite inner product in
Binary_relation
Lattice formed by all integer partitions
lattice is a lattice (and hence also a partially ordered set) Y formed by all integer partitions ordered by inclusion of their Young diagrams (or Ferrers
Young's_lattice
where T0 spaces occur in denotational semantics. The specialization order is also important for identifying suitable topologies on partially ordered sets
Specialization_preorder
Partially ordered set equipped with a rank function
lattice of subspaces of a vector space (dimension of the subspace) Lattice of partitions of a set into finitely many parts, ordered by reverse refinement
Graded_poset
Reversal of the order of elements of a binary relation
Furthermore, the semigroup of endorelations on a set is also a partially ordered structure (with inclusion of relations as sets), and actually an involutive
Converse_relation
Geometric space with four dimensions
everyday life. Single locations in Euclidean 4D space can be given as vectors or 4-tuples, i.e., as ordered lists of numbers such as (x, y, z, w). For example
Four-dimensional_space
order-theoretic mathematics, a series-parallel partial order is a partially ordered set built up from smaller series-parallel partial orders by two simple
Series-parallel_partial_order
Existence of certain infima or suprema of a given poset
assert the existence of certain infima or suprema of a given partially ordered set (poset). The most familiar example is the completeness of the real
Completeness_(order_theory)
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ORDERED VECTOR-SPACE
ORDERED VECTOR-SPACE
ORDERED VECTOR-SPACE
ORDERED VECTOR-SPACE
ORDERED VECTOR-SPACE
ORDERED VECTOR-SPACE
ORDERED VECTOR-SPACE
ORDERED VECTOR-SPACE
ORDERED VECTOR-SPACE
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