Concept
Barrelled space
Related concepts (27)
Mackey space
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. They are named after George Mackey. Examples of locally convex spaces that are Mackey spaces include: All barrelled spaces and more generally all infrabarreled spaces Hence in particular all bornological spaces and reflexive spaces All metrizable spaces.
F-space
In functional analysis, an F-space is a vector space over the real or complex numbers together with a metric such that Scalar multiplication in is continuous with respect to and the standard metric on or Addition in is continuous with respect to The metric is translation-invariant; that is, for all The metric space is complete. The operation is called an F-norm, although in general an F-norm is not required to be homogeneous. By translation-invariance, the metric is recoverable from the F-norm.
Open mapping theorem (functional analysis)
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result which states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map. This proof uses the , and completeness of both and is essential to the theorem. The statement of the theorem is no longer true if either space is just assumed to be a normed space, but is true if and are taken to be Fréchet spaces.
Montel space
In functional analysis and related areas of mathematics, a Montel space, named after Paul Montel, is any topological vector space (TVS) in which an analog of Montel's theorem holds. Specifically, a Montel space is a barrelled topological vector space in which every closed and bounded subset is compact. A topological vector space (TVS) has the if every closed and bounded subset is compact. A is a barrelled topological vector space with the Heine–Borel property.
Semi-reflexive space
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 of X) is bijective. If this map is also an isomorphism of TVSs then it is called reflexive. Semi-reflexive spaces play an important role in the general theory of locally convex TVSs. Since a normable TVS is semi-reflexive if and only if it is reflexive, the concept of semi-reflexivity is primarily used with TVSs that are not normable.
Absorbing set
In functional analysis and related areas of mathematics an absorbing set in a vector space is a set which can be "inflated" or "scaled up" to eventually always include any given point of the vector space. Alternative terms are radial or absorbent set. Every neighborhood of the origin in every topological vector space is an absorbing subset.
Quasi-complete space
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. Every quasi-complete TVS is sequentially complete. In a quasi-complete locally convex space, the closure of the convex hull of a compact subset is again compact. In a quasi-complete Hausdorff TVS, every precompact subset is relatively compact.
Bilinear map
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Let and be three vector spaces over the same base field . A bilinear map is a function such that for all , the map is a linear map from to and for all , the map is a linear map from to In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when we hold the second entry fixed.
Polar set
In functional and convex analysis, and related disciplines of mathematics, the polar set is a special convex set associated to any subset of a vector space lying in the dual space The bipolar of a subset is the polar of but lies in (not ). There are at least three competing definitions of the polar of a set, originating in projective geometry and convex analysis. In each case, the definition describes a duality between certain subsets of a pairing of vector spaces over the real or complex numbers ( and are often topological vector spaces (TVSs)).
Quasibarrelled space
In functional analysis and related areas of mathematics, quasibarrelled spaces are topological vector spaces (TVS) for which every bornivorous barrelled set in the space is a neighbourhood of the origin. Quasibarrelled spaces are studied because they are a weakening of the defining condition of barrelled spaces, for which a form of the Banach–Steinhaus theorem holds. A subset of a topological vector space (TVS) is called bornivorous if it absorbs all bounded subsets of ; that is, if for each bounded subset of there exists some scalar such that A barrelled set or a barrel in a TVS is a set which is convex, balanced, absorbing and closed.