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Operator topologies information


In the mathematical field of functional analysis there are several standard topologies which are given to the algebra B(X) of bounded linear operators on a Banach space X.

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Operator topologies

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functional analysis there are several standard topologies which are given to the algebra B(X) of bounded linear operators on a Banach space X. Let ( T n ) n ∈ N...

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Weak operator topology

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common topologies on B ( H ) {\displaystyle B(H)} , the bounded operators on a Hilbert space H {\displaystyle H} . The strong operator topology, or SOT...

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Strong operator topology

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mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space H induced...

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Weak topology

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mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance...

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Ultraweak topology

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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...

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List of topologies

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The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics....

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Topologies on spaces of linear maps

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variety of topologies. Studying space of linear maps and these topologies can give insight into the spaces themselves. The article operator topologies discusses...

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Von Neumann bicommutant theorem

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topologies on the space of bounded operators, and one can ask what are the *-algebras closed in these topologies. If M is closed in the norm topology...

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Operator algebra

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mechanics – Formulation of quantum mechanics Topologies on the set of operators on a Hilbert space Vertex operator algebra – Algebra used in 2D conformal field...

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Strong topology

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a strong topology is a topology which is stronger than some other "default" topology. This term is used to describe different topologies depending on...

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Comparison of topologies

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possible topologies. See topologies on the set of operators on a Hilbert space for some intricate relationships. All possible polar topologies on a dual...

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Von Neumann algebra

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many other common topologies including the strong, ultrastrong or ultraweak operator topologies. The *-algebras of bounded operators that are closed in...

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Alexandrov topology

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open sets is open; in Alexandrov topologies the finite restriction is dropped. A set together with an Alexandrov topology is known as an Alexandrov-discrete...

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Spectral theory of compact operators

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space H, the compact operators are the closure of the finite rank operators in the uniform operator topology. In general, operators on infinite-dimensional...

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Ultrastrong topology

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set the strong operator and ultrastrong topologies are the same. The ultrastrong topology is stronger than the strong operator topology. One problem with...

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List of functional analysis topics

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density Topologies on the set of operators on a Hilbert space norm topology ultrastrong topology strong operator topology weak operator topology weak-star...

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Operator theory

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linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function...

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Strong convergence

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(as opposed to weak convergence). The convergence of operators in the strong operator topology. This disambiguation page lists articles associated with...

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Topological space

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\}\cup \Gamma } is a topology on X . {\displaystyle X.} Many sets of linear operators in functional analysis are endowed with topologies that are defined...

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Operator norm

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redirect targets Operator algebra – Branch of functional analysis Operator theory – Mathematical field of study Topologies on the set of operators on a Hilbert...

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Kuratowski closure axioms

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any induced topology (relative to the subset A) the closed sets induce a new closure operator that is just the original closure operator restricted to...

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Ergodic theory

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unitary operators on H. Then the operator 1 T ∫ 0 T U t d t {\displaystyle {\frac {1}{T}}\int _{0}^{T}U_{t}\,dt} converges in the strong operator topology as...

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Hilbert space

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with a topology defined by convex open sets Operator theory – Mathematical field of study Operator topologies – Topologies on the set of operators on a...

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General topology

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generates the topology T. Bases are useful because many properties of topologies can be reduced to statements about a base that generates that topology—and because...

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Operator product expansion

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In quantum field theory, the operator product expansion (OPE) is used as an axiom to define the product of fields as a sum over the same fields. As an...

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Metrizable space

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Hilbert space H {\displaystyle {\mathcal {H}}} endowed with the strong operator topology is metrizable (see Proposition II.1 in ). Examples of non-metrizable...

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Abelian von Neumann algebra

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preserve the weak operator topology. As it turns out (and follows easily from the definitions), for algebras L∞(X, μ), the following topologies agree on norm...

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Jordan operator algebra

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functionals continuous for the weak or strong topologies. As a consequence the weak and strong topologies coincide on a JBW algebra. In a JBW algebra,...

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