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Bounded sets are a natural way to define locally convex polar topologies on the vector spaces in a dual pair, as the polar set of a bounded set is an absolutely convex and absorbing set. The concept was first introduced by John von Neumann and Andrey Kolmogorov in 1935.
The von Neumann algebra generated by a set of bounded operators on a Hilbert space is the smallest von Neumann algebra containing all those operators. The tensor product of two von Neumann algebras acting on two Hilbert spaces is defined to be the von Neumann algebra generated by their algebraic tensor product, considered as operators on the ...
A major contribution von Neumann made to measure theory was the result of a paper written to answer a question of Haar regarding whether there existed an algebra of all bounded functions on the real number line such that they form "a complete system of representatives of the classes of almost everywhere-equal measurable bounded functions". [117]
A subset of a TVS is called bounded (or more precisely, von Neumann bounded) if every neighborhood of the origin absorbs it. In a normed space (and even in a seminormed space), a subset is von Neumann bounded if and only if it is norm bounded. Hence, for normed spaces, the notion of a von Neumann bounded set is identical to the usual notion of ...
In topological vector spaces, a different definition for bounded sets exists which is sometimes called von Neumann boundedness. If the topology of the topological vector space is induced by a metric which is homogeneous , as in the case of a metric induced by the norm of normed vector spaces , then the two definitions coincide.
Bound states in the continuum were first predicted in 1929 by Eugene Wigner and John von Neumann. [4] Two potentials were described, in which BICs appear for two different reasons. In this work, a spherically symmetric wave function is first chosen so as to be quadratically integrable over the entire space.
Von Neumann bicommutant theorem. Let M be an algebra consisting of bounded operators on a Hilbert space H, containing the identity operator, and closed under taking adjoints. Then the closures of M in the weak operator topology and the strong operator topology are equal, and are in turn equal to the bicommutant M′′ of M.
The Dirac–von Neumann axioms can be formulated in terms of a C*-algebra as follows.. The bounded observables of the quantum mechanical system are defined to be the self-adjoint elements of the C*-algebra.