In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an ordered vector space or partially ordered vector space is a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
equipped with a
partial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
that is compatible with the vector space operations.
Definition
Given a vector space ''X'' over the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s R and a
preorder
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special cas ...
≤ on the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
''X'', the pair is called a preordered vector space and we say that the preorder ≤ is compatible with the vector space structure of ''X'' and call ≤ a vector preorder on ''X'' if for all ''x'', ''y'', ''z'' in ''X'' and ''λ'' in R with the following two axioms are satisfied
# implies
# implies .
If ≤ is a
partial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
compatible with the vector space structure of ''X'' then is called an ordered vector space and ≤ is called a vector partial order on ''X''.
The two axioms imply that
translation
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
s and
positive homotheties are
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...
s of the order structure and the mapping is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the
dual order structure. Ordered vector spaces are
ordered group
In abstract algebra, a partially ordered group is a group (''G'', +) equipped with a partial order "≤" that is ''translation-invariant''; in other words, "≤" has the property that, for all ''a'', ''b'', and ''g'' in ''G'', if ''a'' ≤ ''b'' ...
s under their addition operation.
Note that ''x'' ≤ ''y'' if and only if −''y'' ≤ −''x''.
Positive cones and their equivalence to orderings
A
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
''C'' of a vector space ''X'' is called a cone if for all real ''r'' > 0, ''rC'' ⊆ ''C''. A cone is called pointed if it contains the origin. A cone ''C'' is convex if and only if ''C'' + ''C'' ⊆ ''C''. The
intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
of any
non-empty family of cones (resp. convex cones) is again a cone (resp. convex cone);
the same is true of the
union of an increasing (under
set inclusion
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
) family of cones (resp. convex cones). A cone ''C'' in a vector space ''X'' is said to be generating if ''X'' = ''C'' − ''C''.
A positive cone is generating if and only if it is a
directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
under ≤.
Given a preordered vector space ''X'', the subset ''X''
+ of all elements ''x'' in (''X'', ≤) satisfying ''x'' ≥ 0 is a pointed
convex cone
In linear algebra, a ''cone''—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under scalar multiplication; that is, is a cone if x\in C implies sx\in C for every .
W ...
with vertex 0 (i.e. it contains 0) called the positive cone of ''X'' and denoted by
.
The elements of the positive cone are called positive.
If ''x'' and ''y'' are elements of a preordered vector space (''X'', ≤), then ''x'' ≤ ''y'' if and only if ''y'' − ''x'' ∈ ''X''
+.
Given any pointed convex cone ''C'' with vertex 0, one may define a preorder ≤ on ''X'' that is compatible with the vector space structure of ''X'' by declaring for all ''x'' and ''y'' in ''X'', that ''x'' ≤ ''y'' if and only if ''y'' − ''x'' ∈ ''C'';
the positive cone of this resulting preordered vector space is ''C''.
There is thus a one-to-one correspondence between pointed convex cones with vertex 0 and vector preorders on ''X''.
If ''X'' is preordered then we may form an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
on ''X'' by defining ''x'' is equivalent to ''y'' if and only if ''x'' ≤ ''y'' and ''y'' ≤ ''x'';
if ''N'' is the
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
containing the origin then ''N'' is a vector subspace of ''X'' and ''X''/''N'' is an ordered vector space under the relation: ''A'' ≤ ''B'' if and only there exist ''a'' in ''A'' and ''b'' in ''B'' such that ''a'' ≤ ''b''.
A subset of ''C'' of a vector space ''X'' is called a
proper cone if it is a convex cone of vertex 0 satisfying ''C'' ∩ (−''C'') = .
Explicitly, ''C'' is a proper cone if (1) ''C'' + ''C'' ⊆ ''C'', (2) ''rC'' ⊆ ''C'' for all ''r'' > 0, and (3) ''C'' ∩ (−''C'') = .
The intersection of any non-empty family of proper cones is again a proper cone. Each proper cone ''C'' in a real vector space induces an order on the vector space by defining ''x'' ≤ ''y'' if and only if ''y'' − ''x'' ∈ ''C'', and furthermore, the positive cone of this ordered vector space will be ''C''. Therefore, there exists a one-to-one correspondence between the proper convex cones of ''X'' and the vector partial orders on ''X''.
By a total vector ordering on ''X'' we mean a
total order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive ...
on ''X'' that is compatible with the vector space structure of ''X''.
The family of total vector orderings on a vector space ''X'' is in one-to-one correspondence with the family of all proper cones that are maximal under set inclusion.
A total vector ordering ''cannot'' be
Archimedean if its
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
, when considered as a vector space over the reals, is greater than 1.
If ''R'' and ''S'' are two orderings of a vector space with positive cones ''P'' and ''Q'', respectively, then we say that ''R'' is finer than ''S'' if ''P'' ⊆ ''Q''.
Examples
The real numbers with the usual ordering form a totally ordered vector space. For all
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s ''n'' ≥ 0, the
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
R
''n'' considered as a vector space over the reals with the
lexicographic order
In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a ...
ing forms a preordered vector space whose order is
Archimedean if and only if ''n'' = 0 or 1.
Pointwise order
If ''S'' is any set and if ''X'' is a vector space (over the reals) of real-valued
functions on ''S'', then the pointwise order on ''X'' is given by, for all ''f'', ''g'' ∈ ''X'', ''f'' ≤ ''g'' if and only if ''f''(''s'') ≤ ''g''(''s'') for all ''s'' in ''S''.
Spaces that are typically assigned this order include:
* the space 𝓁
∞(''S'', R) of
bounded real-valued maps on ''S''.
* the space ''c''
0(R) of real-valued
sequences
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...
that
converge
Converge may refer to:
* Converge (band), American hardcore punk band
* Converge (Baptist denomination), American national evangelical Baptist body
* Limit (mathematics)
* Converge ICT, internet service provider in the Philippines
*CONVERGE CFD s ...
to 0.
* the space ''C''(''S'', R) of
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
real-valued functions on a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
''S''.
* for any non-negative integer ''n'', the Euclidean space R
''n'' when considered as the space ''C''(, R) where ''S'' = is given the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest top ...
.
The space
of all
measurable
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
almost-everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion t ...
bounded real-valued maps on R, where the preorder is defined for all ''f'', ''g'' ∈
by ''f'' ≤ ''g'' if and only if ''f''(''s'') ≤ ''g''(''s'') almost everywhere.
Intervals and the order bound dual
An order interval in a preordered vector space is set of the form
:
'a'', ''b''= ,
:
'a'', ''b''[_=_,
:'a'', ''b''.html" ;"title="=_,
:.html" ;"title="'a'', ''b''[ = ,
:">'a'', ''b''[ = ,
:'a'', ''b''">=_,
:.html" ;"title="'a'', ''b''[ = ,
:">'a'', ''b''[ = ,
:'a'', ''b''= , or
:]''a'', ''b''[ = .
From axioms 1 and 2 above it follows that ''x'', ''y'' ∈
'a'', ''b''and 0 < λ < 1 implies λ''x'' + (1 − ''λ'')''y'' in
'a'', ''b''
thus these order intervals are convex.
A subset is said to be order bounded if it is contained in some order interval.
In a preordered real vector space, if for ''x'' ≥ 0 then the interval of the form
��''x'', ''x'' is
balanced.
An
order unit of a preordered vector space is any element ''x'' such that the set
��''x'', ''x''is
absorbing.
The set of all
linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the ...
s on a preordered vector space ''X'' that map every order interval into a bounded set is called the
order bound dual of ''X'' and denoted by ''X''
''b''.
If a space is ordered then its order bound dual is a vector subspace of its
algebraic dual.
A subset ''A'' of an ordered vector space ''X'' is called
order complete if for every non-empty subset ''B'' ⊆ ''A'' such that ''B'' is order bounded in ''A'', both
and
exist and are elements of ''A''. We say that an ordered vector space ''X'' is order complete is ''X'' is an order complete subset of ''X''.
Examples
If (''X'', ≤) is a preordered vector space over the reals with order unit ''u'', then the map
is a
sublinear functional In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space X is a real-valued function with only some of the properties of a seminorm. ...
.
Properties
If ''X'' is a preordered vector space then for all ''x'', ''y'' ∈ ''X'',
* ''x'' ≥ 0 and ''y'' ≥ 0 imply ''x'' + ''y'' ≥ 0.
* ''x'' ≤ ''y'' if and only if −''y'' ≤ −''x''.
* ''x'' ≤ ''y'' and ''r'' < 0 imply ''rx'' ≥ ''ry''.
* ''x'' ≤ ''y'' if and only if ''y'' = sup if and only if ''x'' = inf.
* sup exists if and only if inf exists, in which case inf = −sup.
* sup exists if and only if inf exists, in which case for all ''z'' ∈ ''X'',
** sup = ''z'' + sup, and
** inf = ''z'' + inf
** ''x'' + ''y'' = inf + sup.
* ''X'' is a
vector lattice
In mathematics, a 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 named after Frigyes Riesz who first defined them in his 1928 paper ''S ...
if and only if sup exists for all ''x'' in ''X''.
Spaces of linear maps
A cone
is said to be generating if
is equal to the whole vector space.
If
and
are two non-trivial ordered vector spaces with respective positive cones
and
then
is generating in
if and only if the set
is a proper cone in
which is the space of all linear maps from
into
In this case, the ordering defined by
is called the canonical ordering of
More generally, if
is any vector subspace of
such that
is a proper cone, the ordering defined by
is called the canonical ordering of
Positive functionals and the order dual
A
linear function
In mathematics, the term linear function refers to two distinct but related notions:
* In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For dist ...
on a preordered vector space is called positive if it satisfies either of the following equivalent conditions:
#
implies
# if
then
The set of all positive linear forms on a vector space with positive cone
called the
dual cone
Dual or Duals may refer to:
Paired/two things
* Dual (mathematics), a notion of paired concepts that mirror one another
** Dual (category theory), a formalization of mathematical duality
*** see more cases in :Duality theories
* Dual (grammatica ...
and denoted by
is a cone equal to the
polar
Polar may refer to:
Geography
Polar may refer to:
* Geographical pole, either of two fixed points on the surface of a rotating body or planet, at 90 degrees from the equator, based on the axis around which a body rotates
*Polar climate, the cli ...
of
The preorder induced by the dual cone on the space of linear functionals on
is called the .
The
order dual of an ordered vector space
is the set, denoted by
defined by
Although
there do exist ordered vector spaces for which set equality does hold.
Special types of ordered vector spaces
Let ''X'' be an ordered vector space. We say that an ordered vector space ''X'' is
Archimedean ordered
In mathematics, specifically in order theory, a binary relation \,\leq\, on a vector space X over the real or complex numbers is called Archimedean if for all x \in X, whenever there exists some y \in X such that n x \leq y for all positive inte ...
and that the order of ''X'' is Archimedean if whenever ''x'' in ''X'' is such that
is
majorized (i.e. there exists some ''y'' in ''X'' such that ''nx'' ≤ ''y'' for all
) then .
A
topological vector space
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 investigated in functional analysis.
A topological vector space is a vector space that is als ...
(TVS) that is an ordered vector space is necessarily Archimedean if its positive cone is closed.
We say that a preordered vector space ''X'' is
regularly ordered
In mathematics, specifically in order theory and functional analysis, an ordered vector space X is said to be regularly ordered and its order is called regular if X is Archimedean ordered and the order dual of X distinguishes points in X.
...
and that its order is regular if it is
Archimedean ordered
In mathematics, specifically in order theory, a binary relation \,\leq\, on a vector space X over the real or complex numbers is called Archimedean if for all x \in X, whenever there exists some y \in X such that n x \leq y for all positive inte ...
and ''X''
+ distinguishes points in ''X''.
This property guarantees that there are sufficiently many positive linear forms to be able to successfully use the tools of duality to study ordered vector spaces.
An ordered vector space is called a
vector lattice
In mathematics, a 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 named after Frigyes Riesz who first defined them in his 1928 paper ''S ...
if for all elements ''x'' and ''y'', the
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...
sup(''x'', ''y'') and
infimum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
inf(''x'', ''y'') exist.
Subspaces, quotients, and products
Throughout let ''X'' be a preordered vector space with positive cone ''C''.
;Subspaces
If ''M'' is a vector subspace of ''X'' then the canonical ordering on ''M'' induced by ''Xs positive cone ''C'' is the partial order induced by the pointed convex cone ''C'' ∩ ''M'', where this cone is proper if ''C'' is proper.
;Quotient space
Let ''M'' be a vector subspace of an ordered vector space ''X'',
be the canonical projection, and let
.
Then
is a cone in ''X''/''M'' that induces a canonical preordering on the
quotient space ''X''/''M''.
If
is a proper cone in ''X''/''M'' then
makes ''X''/''M'' into an ordered vector space.
If ''M'' is
''C''-saturated then
defines the canonical order of ''X''/''M''.
Note that
provides an example of an ordered vector space where
is not a proper cone.
If ''X'' is also a
topological vector space
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 investigated in functional analysis.
A topological vector space is a vector space that is als ...
(TVS) and if for each
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
''V'' of 0 in ''X'' there exists a neighborhood ''U'' of 0 such that
''U'' + ''N'') ∩ C⊆ ''V'' + ''N'' then
is a
normal cone for the
quotient topology
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...
.
If ''X'' is a
topological vector lattice
In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) X that has a partial order \,\leq\, making it into vector lattice that is possesses a neighborhoo ...
and ''M'' is a closed
solid
Solid is one of the four fundamental states of matter (the others being liquid, gas, and plasma). The molecules in a solid are closely packed together and contain the least amount of kinetic energy. A solid is characterized by structur ...
sublattice of ''X'' then ''X''/''L'' is also a topological vector lattice.
;Product
If ''S'' is any set then the space ''X''
''S'' of all functions from ''S'' into ''X'' is canonically ordered by the proper cone
.
Suppose that
is a family of preordered vector spaces and that the positive cone of
is
.
Then
is a pointed convex cone in
, which determines a canonical ordering on
;
''C'' is a proper cone if all
are proper cones.
;Algebraic direct sum
The algebraic
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
of
is a vector subspace of
that is given the canonical subspace ordering inherited from
.
If ''X''
1, ..., ''X''
''n'' are ordered vector subspaces of an ordered vector space ''X'' then ''X'' is the ordered direct sum of these subspaces if the canonical algebraic isomorphism of ''X'' onto
(with the canonical product order) is an
order isomorphism
In the mathematical field of order theory, an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can be cons ...
.
Examples
* The
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s with the usual order is an ordered vector space.
* R
2 is an ordered vector space with the ≤ relation defined in any of the following ways (in order of increasing strength, i.e., decreasing sets of pairs):
**
Lexicographical order
In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of ...
: if and only if or ( and ). This is a
total order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive ...
. The positive cone is given by or ( and ), i.e., in
polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
, the set of points with the angular coordinate satisfying , together with the origin.
** if and only if and (the
product order
In mathematics, given two preordered sets A and B, the product order (also called the coordinatewise orderDavey & Priestley, ''Introduction to Lattices and Order'' (Second Edition), 2002, p. 18 or componentwise order) is a partial ordering o ...
of two copies of R with "≤"). This is a partial order. The positive cone is given by and , i.e., in polar coordinates , together with the origin.
** if and only if ( and ) or ( and ) (the reflexive closure of the
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
of two copies of R with "<"). This is also a partial order. The positive cone is given by ( and ) or (), i.e., in polar coordinates, , together with the origin.
:Only the second order is, as a subset of R
4, closed; see
partial orders in topological spaces.
:For the third order the two-dimensional "
intervals" are
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
s which generate the topology.
* R
''n'' is an ordered vector space with the ≤ relation defined similarly. For example, for the second order mentioned above:
** if and only if for .
* A
Riesz space
In mathematics, a 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 named after Frigyes Riesz who first defined them in his 1928 paper ''Su ...
is an ordered vector space where the order gives rise to a
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
.
* The space of continuous functions on where if and only if for all ''x'' in .
See also
*
*
*
*
*
*
*
*
*
*
References
Bibliography
*
*
Bourbaki, Nicolas
Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook ...
;
Elements of Mathematics: Topological Vector Spaces; .
*
*
*
{{Order theory
Functional analysis
Ordered groups
Vector spaces