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The Neumann series is used in functional analysis. It is closely connected to the resolvent formalism for studying the spectrum of bounded operators and, applied from the left to a function, it forms the Liouville-Neumann series that formally solves Fredholm integral equations.
The collection of all bounded sets on a topological vector space is called the von Neumann bornology or the (canonical) bornology of .. A base or fundamental system of bounded sets of is a set of bounded subsets of such that every bounded subset of is a subset of some . [1] The set of all bounded subsets of trivially forms a fundamental system of bounded sets of .
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]
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 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 ...
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.
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.
In mathematics, particularly in operator theory, Wold decomposition or Wold–von Neumann decomposition, named after Herman Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space. It states that every isometry is a direct sum of copies of the unilateral shift and a unitary operator.