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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.
Given a surjective map a point is called a point of openness for and is said to be open at (or an open map at ) if for every open neighborhood of is a neighborhood of in (note that the neighborhood is not required to be an open neighborhood).
A surjective map is called an open map if it is open at every point of its domain, while it is called an almost open map if each of its fibers has some point of openness. Explicitly, a surjective map is said to be almost open if for every there exists some such that is open at Every almost open surjection is necessarily a pseudo-open map (introduced by Alexander Arhangelskii in 1963), which by definition means that for every and every neighborhood of (that is, ), is necessarily a neighborhood of
A linear map between two topological vector spaces (TVSs) is called a nearly open linear map or an almost open linear map if for any neighborhood of in the closure of in is a neighborhood of the origin. Importantly, some authors use a different definition of "almost open map" in which they instead require that the linear map satisfy: for any neighborhood of in the closure of in (rather than in ) is a neighborhood of the origin; this article will not use this definition. [1]
If a linear map is almost open then because is a vector subspace of that contains a neighborhood of the origin in the map is necessarily surjective. For this reason many authors require surjectivity as part of the definition of "almost open".
If is a bijective linear operator, then is almost open if and only if is almost continuous. [1]
Every surjective open map is an almost open map but in general, the converse is not necessarily true. If a surjection is an almost open map then it will be an open map if it satisfies the following condition (a condition that does not depend in any way on 's topology ):
If the map is continuous then the above condition is also necessary for the map to be open. That is, if is a continuous surjection then it is an open map if and only if it is almost open and it satisfies the above condition.
The two theorems above do not require the surjective linear map to satisfy any topological conditions.
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations are also continuous functions. Such a topology is called a vector topology and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and completeness. Some authors also require that the space is a Hausdorff space. One of the most widely studied categories of TVSs are locally convex topological vector spaces. This article focuses on TVSs that are not necessarily locally convex. Other well-known examples of TVSs include Banach spaces, Hilbert spaces and Sobolev spaces.
In functional analysis, an F-space is a vector space over the real or complex numbers together with a metric such that
In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space for which the canonical evaluation map from into its bidual is a homeomorphism. A normed space is reflexive if and only if this canonical evaluation map is surjective, in which case this evaluation map is an isometric isomorphism and the normed space is a Banach space. Those spaces for which the canonical evaluation map is surjective are called semi-reflexive spaces.
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces.
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function is open if for any open set in the image is open in Likewise, a closed map is a function that maps closed sets to closed sets. A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa.
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 functional analysis and related areas of mathematics, a barrelled 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 barrel in a topological vector space is a set that is convex, balanced, absorbing, and closed. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them. Barrelled spaces were introduced by Bourbaki.
In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem, is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.
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 mathematics, particularly in functional analysis and topology, closed graph is a property of functions. A function f : X → Y between topological spaces has a closed graph if its graph is a closed subset of the product space X × Y. A related property is open graph.
In the branch of mathematics called functional analysis, a complemented subspace of a topological vector space is a vector subspace for which there exists some other vector subspace of called its (topological) complement in , such that is the direct sum in the category of topological vector spaces. Formally, topological direct sums strengthen the algebraic direct sum by requiring certain maps be continuous; the result retains many nice properties from the operation of direct sum in finite-dimensional vector spaces.
The theorem on the surjection of Fréchet spaces is an important theorem, due to Stefan Banach, that characterizes when a continuous linear operator between Fréchet spaces is surjective.
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 functional analysis, a topological vector space (TVS) is called ultrabornological if every bounded linear operator from into another TVS is necessarily continuous. A general version of the closed graph theorem holds for ultrabornological spaces. Ultrabornological spaces were introduced by Alexander Grothendieck.
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.
This is a glossary for the terminology in a mathematical field of functional analysis.
In functional analysis and related areas of mathematics, a metrizable topological vector space (TVS) is a TVS whose topology is induced by a metric. An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.
F. Riesz's theorem is an important theorem in functional analysis that states that a Hausdorff topological vector space (TVS) is finite-dimensional if and only if it is locally compact. The theorem and its consequences are used ubiquitously in functional analysis, often used without being explicitly mentioned.
In mathematics, particularly in functional analysis, the closed graph theorem is a result connecting the continuity of a linear operator to a topological property of their graph. Precisely, the theorem states that a linear operator between two Banach spaces is continuous if and only if the graph of the operator is closed.