In mathematics, particularly in functional analysis, a Mackey space is a locally convex topological vector space X such that the topology of X coincides with the Mackey topology τ(X,X′), the finest topology which still preserves the continuous dual.
In mathematics, an LF-space, also written (LF)-space, is a topological vector space (TVS) X that is a locally convex inductive limit of a countable inductive system of Fréchet spaces. This means that X is a direct limit of a direct system in the category of locally convex topological vector spaces and each is a Fréchet space. The name LF stands for Limit of Fréchet spaces.
In mathematics, specifically in order theory and functional analysis, the order dual of an ordered vector space is the set where denotes the set of all positive linear functionals on , where a linear function on is called positive if for all implies The order dual of is denoted by . Along with the related concept of the order bound dual, this space plays an important role in the theory of ordered topological vector spaces.
The Mackey–Arens theorem is an important theorem in functional analysis that characterizes those locally convex vector topologies that have some given space of linear functionals as their continuous dual space. According to Narici (2011), this profound result is central to duality theory; a theory that is "the central part of the modern theory of topological vector spaces."
In the mathematical disciplines of in functional analysis and order theory, a Banach lattice(X,‖·‖) is a complete normed vector space with a lattice order, such that for all x, y ∈ X, the implication
A locally convex topological vector space (TVS) is B-complete or a Ptak space if every subspace is closed in the weak-* topology on whenever is closed in for each equicontinuous subset .
In the field of functional analysis, DF-spaces, also written (DF)-spaces are locally convex topological vector space having a property that is shared by locally convex metrizable topological vector spaces. They play a considerable part in the theory of topological tensor products.
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 whose positive cone is a closed subset of X. Ordered TVS have important applications in spectral theory.
In mathematics, specifically in order theory and functional analysis, a subset of a vector lattice is said to be solid and is called an ideal if for all and if then An ordered vector space whose order is Archimedean is said to be Archimedean ordered. If then the ideal generated by is the smallest ideal in containing An ideal generated by a singleton set is called a principal ideal in
In mathematics, specifically in order theory and functional analysis, a subset of an ordered vector space is said to be order complete in if for every non-empty subset of that is order bounded in , the supremum ' and the infimum both exist and are elements of An ordered vector space is called order complete, Dedekind complete, a complete vector lattice, or a complete Riesz space, if it is order complete as a subset of itself, in which case it is necessarily a vector lattice. An ordered vector space is said to be countably order complete if each countable subset that is bounded above has a supremum.
In mathematics, specifically in order theory and functional analysis, the order topology of an ordered vector space is the finest locally convex topological vector space (TVS) topology on for which every order interval is bounded, where an order interval in is a set of the form where and belong to
In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) that has a partial order making it into vector lattice that is possesses a neighborhood base at the origin consisting of solid sets. Ordered vector lattices have important applications in spectral theory.
In mathematics, specifically in order theory and functional analysis, an element of an ordered topological vector space is called a quasi-interior point of the positive cone of if and if the order interval is a total subset of ; that is, if the linear span of is a dense subset of
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.
In mathematics, specifically in order theory and functional analysis, a Fréchet lattice is a topological vector lattice that is also a Fréchet space. Fréchet lattices are important in the theory of topological vector lattices.
In mathematics, specifically in order theory and functional analysis, a normed lattice is 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 lattices. They are closely related to Banach vector lattices, which are normed vector lattices that are also Banach spaces.
In mathematics, specifically in order theory and functional analysis, if is a cone at 0 in a vector space such that then a subset is said to be -saturated if where Given a subset the -saturated hull of is the smallest -saturated subset of that contains If is a collection of subsets of then
In mathematics, specifically in order theory and functional analysis, a band in a vector lattice is a subspace of that is solid and such that for all such that exists in we have The smallest band containing a subset of is called the band generated by in A band generated by a singleton set is called a principal band.
In mathematics, specifically in order theory and functional analysis, a sequence of positive elements in a preordered vector space is called order summable if exists in . For any , we say that a sequence of positive elements of is of type if there exists some and some sequence in such that for all .
In mathematics, specifically in order theory and functional analysis, a filter in an order complete vector lattice is order convergent if it contains an order bounded subset and if
