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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] |
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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] |
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Polarization Identity
In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product then the polarization identity can be used to express this inner product entirely in terms of the norm. The polarization identity shows that a norm can arise from at most one inner product; however, there exist norms that do not arise from any inner product. The norm associated with any inner product space satisfies the parallelogram law: \, x+y\, ^2 + \, x-y\, ^2 = 2\, x\, ^2 + 2\, y\, ^2. In fact, as observed by John von Neumann, the parallelogram law characterizes those norms that arise from inner products. Given a normed space (H, \, \cdot\, ), the parallelogram law holds for \, \cdot\, if and only if there exists an inner product \langle \cdot, \cdot \rangle on H such that \, x\, ^2 = \langle x,\ x\rangle for all x \in H, in whi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Base (topology)
In mathematics, a base (or basis; : bases) for the topology of a topological space is a family \mathcal of open subsets of such that every open set of the topology is equal to the union of some sub-family of \mathcal. For example, the set of all open intervals in the real number line \R is a basis for the Euclidean topology on \R because every open interval is an open set, and also every open subset of \R can be written as a union of some family of open intervals. Bases are ubiquitous throughout topology. The sets in a base for a topology, which are called , are often easier to describe and use than arbitrary open sets. Many important topological definitions such as continuity and convergence can be checked using only basic open sets instead of arbitrary open sets. Some topologies have a base of open sets with specific useful properties that may make checking such topological definitions easier. Not all families of subsets of a set X form a base for a topology on X. Under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Norm (mathematics)
In mathematics, a norm is a function (mathematics), function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the Origin (mathematics), origin: it Equivariant map, commutes with scaling, obeys a form of the triangle inequality, and zero is only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a norm on the associated Euclidean vector space, called the #Euclidean norm, Euclidean norm, the #p-norm, 2-norm, or, sometimes, the magnitude or length of the vector. This norm can be defined as the square root of the inner product of a vector with itself. A seminorm satisfies the first two properties of a norm but may be zero for vectors other than the origin. A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a ''seminormed vector space''. The term pseudonorm has been used for several related meaning ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Normed Vector Space
The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898. The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war period. It was badly damaged during World War II (1939–45). In the first thirty years after the war the shipyard again experienced a boom and employed up to 3,000 workers making oil tankers, and then liquid natural gas tankers. Demand dropped off in the 1970s and 1980s. In 1972 the shipyard became Chantiers de France-Dunkerque, and in 1983 merged with others yards to become part of Chantiers du Nord et de la Mediterranee, or Normed. The shipyard closed in 1987. Foundation (1898–99) The Ateliers et Chantiers de France (ACF) company was officially founded on 6 July 1898 by a consortium of six shipping brokers, the Dunkirk chamber of commerce and the state. The state asked that the shipyard be able to build steamships and also four-maste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Banach–Alaoglu Theorem
In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak-* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact. This theorem has applications in physics when one describes the set of states of an algebra of observables, namely that any state can be written as a convex linear combination of so-called pure states. History According to Lawrence Narici and Edward Beckenstein, the Alaoglu theorem is a “very important result—maybe most important fact about the weak-* topology—hatechos throughout functional analysis.” In 1912, Helly proved that the unit ball of the continuous dual space of C( , b is countably weak-* co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Dominated Convergence Theorem
In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if a sequence of functions is bounded in absolute value by an integrable function and is almost everywhere pointwise convergent to a function then the sequence converges in L_1 to its pointwise limit, and in particular the integral of the limit is the limit of the integrals. Its power and utility are two of the primary theoretical advantages of Lebesgue integration over Riemann integration. In addition to its frequent appearance in mathematical analysis and partial differential equations, it is widely used in probability theory, since it gives a sufficient condition for the convergence of expected values of random variables. Statement Lebesgue's dominated convergence theorem. Let (f_n) be a sequence of complex-valued measurable functions on a measure space . Suppose that t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Finite-rank Operator
In functional analysis, a branch of mathematics, a finite-rank operator is a bounded linear operator between Banach spaces whose range is finite-dimensional. Finite-rank operators on a Hilbert space A canonical form Finite-rank operators are matrices (of finite size) transplanted to the infinite dimensional setting. As such, these operators may be described via linear algebra techniques. From linear algebra, we know that a rectangular matrix, with complex entries, M \in \mathbb^ has rank 1 if and only if M is of the form :M = \alpha \cdot u v^*, \quad \mbox \quad \, u \, = \, v\, = 1 \quad \mbox \quad \alpha \geq 0 . The same argument and Riesz' lemma show that an operator T on a Hilbert space H is of rank 1 if and only if :T h = \alpha \langle h, v\rangle u \quad \mbox \quad h \in H , where the conditions on \alpha, u, v are the same as in the finite dimensional case. Therefore, by induction, an operator T of finite rank n takes the form :T h = \sum _ ^n \alpha_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Weak-star Operator 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] |
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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] |
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Convex Set
In geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary (topology), boundary of a convex set in the plane is always a convex curve. The intersection of all the convex sets that contain a given subset of Euclidean space is called the convex hull of . It is the smallest convex set containing . A convex function is a real-valued function defined on an interval (mathematics), interval with the property that its epigraph (mathematics), epigraph (the set of points on or above the graph of a function, graph of the function) is a convex set. Convex minimization is a subfield of mathematical optimization, optimization that studies the problem of minimizing convex functions over convex sets. The branch of mathematics devoted to the study of properties of convex sets and convex f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |