In mathematics, more specifically in functional analysis, a subset of a topological vector space is said to be a total subset of if the linear span of is a dense subset of [1] This condition arises frequently in many theorems of functional analysis.
In mathematics, more specifically in functional analysis, a subset of a topological vector space is said to be a total subset of if the linear span of is a dense subset of [1] This condition arises frequently in many theorems of functional analysis.
Unbounded self-adjoint operators on Hilbert spaces are defined on total subsets.
In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous.
In mathematics, particularly in functional analysis, a webbed space is a topological vector space designed with the goal of allowing the results of the open mapping theorem and the closed graph theorem to hold for a wider class of linear maps whose codomains are webbed spaces. A space is called webbed if there exists a collection of sets, called a web that satisfies certain properties. Webs were first investigated by de Wilde.
In functional analysis, a total set in a vector space is a set of linear functionals with the property that if a vector satisfies for all then is the zero vector.
A locally convex topological vector space (TVS) Failed to parse : X 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 functional analysis, a topological vector space (TVS) is said to be quasi-complete or boundedly complete if every closed and bounded subset is complete. This concept is of considerable importance for non-metrizable TVSs.
In functional analysis, a topological homomorphism or simply homomorphism is the analog of homomorphisms for the category of topological vector spaces (TVSs). This concept is of considerable importance in functional analysis and the famous open mapping theorem gives a sufficient condition for a continuous linear map between Fréchet spaces to be a topological homomorphism.
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, an element of a vector lattice is called a weak order unit in if and also for all
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, an abstract L-space, an AL-space, or an abstract Lebesgue space is a Banach lattice whose norm is additive on the positive cone of X.
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 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 functional analysis, a family of subsets a topological vector space (TVS) is said to be saturated if contains a non-empty subset of and if for every the following conditions all hold:
In mathematics, more specifically in functional analysis, a positive linear operator from an preordered vector space into a preordered vector space is a linear operator on into such that for all positive elements of that is it holds that In other words, a positive linear operator maps the positive cone of the domain into the positive cone of the codomain.
In functional analysis, a subset of a real or complex vector space that has an associated vector bornology is called bornivorous and a bornivore if it absorbs every element of If is a topological vector space (TVS) then a subset of is bornivorous if it is bornivorous with respect to the von-Neumann bornology of .
In functional analysis and related areas of mathematics, an almost open map between topological spaces is a map that satisfies a condition similar to, but weaker than, the condition of being an open map. As described below, for certain broad categories of topological vector spaces, all surjective linear operators are necessarily almost open.
| Spaces |
| ||||
|---|---|---|---|---|---|
| Theorems | |||||
| Operators | |||||
| Algebras | |||||
| Open problems | |||||
| Applications | |||||
| Advanced topics | |||||
 | This linear algebra-related article is a stub. You can help Wikipedia by expanding it. |
