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Ultraweak Topology
In functional analysis, a branch of mathematics, the ultraweak topology, also called the weak-* topology, or weak-* operator topology or σ-weak topology, is a topology on ''B''(''H''), the space of bounded operators on a Hilbert space ''H''. ''B''(''H'') admits a predual ''B''*(''H''), the trace class operators on ''H''. The ultraweak topology is the weak-* topology so induced; in other words, the ultraweak topology is the weakest topology such that predual elements remain continuous on ''B''(''H''). Relation with the weak (operator) topology The ultraweak topology is similar to the weak operator topology. For example, on any norm-bounded set the weak operator and ultraweak topologies are the same, and in particular, the unit ball is compact in both topologies. The ultraweak topology is stronger than the weak operator topology. One problem with the weak operator topology is that the dual of ''B''(''H'') with the weak operator topology is "too small". The ultraweak topology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 (for example, Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, or Topological space#Definitions, topology) and the linear transformation, linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous function, continuous or unitary operator, unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a topological space is a Set (mathematics), set whose elements are called Point (geometry), points, along with an additional structure called a topology, which can be defined as a set of Neighbourhood (mathematics), neighbourhoods for each point that satisfy some Axiom#Non-logical axioms, axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a space (mathematics), mathematical space that allows for the definition of Limit (mathematics), limits, Continuous function (topology), continuity, and Connected space, connectedness. Common types ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded Operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector spaces (a special type of TVS), then L is bounded if and only if there exists some M > 0 such that for all x \in X, \, Lx\, _Y \leq M \, x\, _X. The smallest such M is called the operator norm of L and denoted by \, L\, . A linear operator between normed spaces is continuous if and only if it is bounded. The concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces. Outside of functional analysis, when a function f : X \to Y is called " bounded" then this usually means that its image f(X) is a bounded subset of its codomain. A linear map has this property if and only if it is identically 0. Consequently, in functional analysis, when a linear operator is called "bounded" then it is never ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The inner product allows lengths and angles to be defined. Furthermore, Complete metric space, completeness means that there are enough limit (mathematics), limits in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of a Banach space. Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, mathematical formulation of quantum mechanics, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Predual
In mathematics, the predual of an object ''D'' is an object ''P'' whose dual space is ''D''. For example, the predual of the space of bounded operators is the space of trace class In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of tra ... operators, and the predual of the space ''L''∞(R) of essentially bounded functions on R is the Banach space ''L''1(R) of integrable functions. Abstract algebra Functional analysis {{mathanalysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace Class
In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace-class operators generalizes the trace of matrices studied in linear algebra. All trace-class operators are compact operators. In quantum mechanics, quantum states are described by density matrices, which are certain trace class operators. Trace-class operators are essentially the same as nuclear operators, though many authors reserve the term "trace-class operator" for the special case of nuclear operators on Hilbert spaces and use the term "nuclear operator" in more general topological vector spaces (such as Banach spaces). Definition Let H be a separable Hilbert space, \left\_^ an orthonormal basis and A : H \to H a positive bounded linear operator on H. The trace of A is denoted by \operatorname (A) and defined as :\operato ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space (such as a normed vector space) with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis. One may call subsets of a topological vector space weakly closed (respectively, weakly compact, etc.) if they are closed (respectively, compact, etc.) with respect to the weak topology. Likewise, functions are sometimes called weakly continuous (respectively, weakly differentiable, weakly analytic, etc.) if they are continuous (respectively, differentiable, analytic, etc.) with respect to the weak topology. History Starting in the early 1900s, David Hilbert and Marcel Riesz made extensive use of weak convergence. The early pioneers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topologies On The Set Of Operators On A Hilbert Space
In the mathematical field of functional analysis there are several standard topologies which are given to the algebra of bounded linear operators on a Banach space . Introduction Let (T_n)_ be a sequence of linear operators on the Banach space X. Consider the statement that (T_n)_ converges to some operator T on X. This could have several different meanings: * If \, T_n - T\, \to 0, that is, the operator norm of T_n - T (the supremum of \, T_n x - T x \, _X, where x ranges over the unit ball in X) converges to 0, we say that T_n \to T in the uniform operator topology. * If T_n x \to Tx for all x \in X, then we say T_n \to T in the strong operator topology. * Finally, suppose that for all x \in X we have T_n x \to Tx in the weak topology of X. This means that F(T_n x) \to F(T x) for all continuous linear functionals F on X. In this case we say that T_n \to T in the weak operator topology. List of topologies on B(''H'') There are many topologies that can be defined o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Operator Topology
In functional analysis, the weak operator topology, often abbreviated WOT,Ilijas Farah, Combinatorial Set Theory of C*-algebras' (2019), p. 80. is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional sending an operator T to the complex number \langle Tx, y\rangle is continuous for any vectors x and y in the Hilbert space. Explicitly, for an operator T there is base of neighborhoods of the following type: choose a finite number of vectors x_i, continuous functionals y_i, and positive real constants \varepsilon_i indexed by the same finite set I. An operator S lies in the neighborhood if and only if , y_i(T(x_i) - S(x_i)), 0. Relationships between different topologies on ''B(X,Y)'' The different terminology for the various topologies on B(X,Y) can sometimes be confusing. For instance, "strong convergence" for vectors in a normed space sometimes refers to norm-convergence, which is very often distinct from (and stronger than) tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology Of Function Spaces
Topology (from the Greek words , and ) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set endowed with a structure, called a ''topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. The following are basic examples of topological properties: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a lin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
