In mathematics, specifically in
order theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
and
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
, a locally convex vector lattice (LCVL) 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 neighborhood ba ...
that is also a
locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...
space.
LCVLs are important in the theory of
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 neighborhood ba ...
s.
Lattice semi-norms
The
Minkowski functional
In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space.
If K is a subset of a real or complex vector space X, then ...
of a convex,
absorbing, and
solid
Solid is one of the State of matter#Four fundamental states, four fundamental states of matter (the others being liquid, gas, and Plasma (physics), plasma). The molecules in a solid are closely packed together and contain the least amount o ...
set is a called a lattice semi-norm.
Equivalently, it is a
semi-norm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
such that
implies
The topology of a locally convex vector lattice is generated by the family of all continuous lattice semi-norms.
Properties
Every locally convex vector lattice possesses a neighborhood base at the origin consisting of
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
balanced
In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ...
solid
Solid is one of the State of matter#Four fundamental states, four fundamental states of matter (the others being liquid, gas, and Plasma (physics), plasma). The molecules in a solid are closely packed together and contain the least amount o ...
absorbing sets.
The strong dual of a locally convex vector lattice
is an
order complete In mathematics, specifically in order theory and functional analysis, a subset A of an ordered vector space is said to be order complete in X if for every non-empty subset S of C that is order bounded in A (meaning contained in an interval, which is ...
locally convex vector lattice (under its canonical order) and it is a
solid
Solid is one of the State of matter#Four fundamental states, four fundamental states of matter (the others being liquid, gas, and Plasma (physics), plasma). The molecules in a solid are closely packed together and contain the least amount o ...
subspace of the
order dual of
;
moreover, if
is a
barreled space
In functional analysis and related areas of 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 for the zero vector.
A barrelled set or a ...
then the continuous dual space of
is a band in the order dual of
and the strong dual of
is a complete locally convex TVS.
If a locally convex vector lattice is
barreled then its strong dual space is complete (this is not necessarily true if the space is merely a locally convex barreled space but not a locally convex vector lattice).
If a locally convex vector lattice
is
semi-reflexive In the area of 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 ''X'' into its bidual (which is the strong dual of the strong dual ...
then it is order complete and
(that is,
) is a complete TVS;
moreover, if in addition every positive linear functional on
is continuous then
is of
is of
minimal type, the
order topology
In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.
If ''X'' is a totally ordered set, t ...
on
is equal to the
Mackey topology In functional analysis and related areas of mathematics, the Mackey 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 topology does not mak ...
and
is
reflexive.
Every
reflexive locally convex vector lattice is
order complete In mathematics, specifically in order theory and functional analysis, a subset A of an ordered vector space is said to be order complete in X if for every non-empty subset S of C that is order bounded in A (meaning contained in an interval, which is ...
and a complete locally convex TVS whose strong dual is a
barreled reflexive locally convex TVS that can be identified under the canonical evaluation map with the strong bidual (that is, the strong dual of the strong dual).
If a locally convex vector lattice
is an
infrabarreled TVS then it can be identified under the evaluation map with a topological vector sublattice of its strong bidual, which is an order complete locally convex vector lattice under its canonical order.
If
is a
separable metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...
locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...
ordered topological vector space In mathematics, specifically in functional 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 ≤ making it into an ordered vector space whos ...
whose positive cone
is a complete and total subset of
then the set of
quasi-interior points of
is dense in
If
is a locally convex vector lattice that is
bornological and
sequentially complete
In mathematics, specifically in topology and functional analysis, a subspace of a uniform space is said to be sequentially complete or semi-complete if every Cauchy sequence in converges to an element in .
is called sequentially complete if i ...
, then there exists a family of compact spaces
and a family of
-indexed vector lattice embeddings
such that
is the finest locally convex topology on
making each
continuous.
Examples
Every
Banach lattice,
normed lattice, and
Fréchet lattice is a locally convex vector lattice.
See also
*
*
*
*
References
Bibliography
*
*
{{Order theory
Functional analysis