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 normed lattice 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
normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
whose unit ball is a
solid set In mathematics, specifically in order theory and functional analysis, a subset S of a vector lattice is said to be solid and is called an ideal if for all s \in S and x \in X, if , x, \leq , s, then x \in S.
An ordered vector space whose order is ...
.
Normed lattices 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. They are closely related to
Banach vector lattice
Banach (pronounced in German, in Slavic Languages, and or in English) is a Jewish surname of Ashkenazi origin believed to stem from the translation of the phrase " son of man", combining the Hebrew word ''ben'' ("son of") and Arameic ''nasha' ...
s, which are normed vector lattices that are also
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s.
Properties
Every normed lattice is a
locally convex vector lattice In mathematics, specifically in order theory and functional analysis, a locally 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.
La ...
.
The strong dual of a normed lattice is a
Banach lattice In the mathematical disciplines of in functional analysis and order theory, a Banach lattice is a complete normed vector space with a lattice order, such that for all , the implication \Rightarrow holds, where the absolute value is defined as , ...
with respect to the dual norm and canonical order.
If it is also a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
then its continuous dual space is equal to its
order dual.
Examples
Every
Banach lattice In the mathematical disciplines of in functional analysis and order theory, a Banach lattice is a complete normed vector space with a lattice order, such that for all , the implication \Rightarrow holds, where the absolute value is defined as , ...
is a normed lattice.
See also
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References
Bibliography
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{{Order theory
Functional analysis