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Mathematical Foundations of Quantum MechanicsNew Edition$
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John von Neumann

Print publication date: 2018

Print ISBN-13: 9780691178561

Published to Princeton Scholarship Online: September 2018

DOI: 10.23943/princeton/9780691178561.001.0001

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Abstract Hilbert Space

Abstract Hilbert Space

(p.25) Chapter II Abstract Hilbert Space
Mathematical Foundations of Quantum Mechanics

John von Neumann

, Robert T. Beyer, Nicholas A. Wheeler
Princeton University Press

This chapter defines Hilbert space, which furnishes the mathematical basis for the treatment of quantum mechanics. This is done within the context of an equation introduced in the previous chapter, and which have accordingly the same meaning in the “discrete” function space FsubscriptZ of the sequences xsubscriptv (ν‎ = 1, 2, . . .) and in the “continuous” Fsubscript Greek Capital Letter Omega of the wave functions φ‎(q₁, . . . , qₖ) (q₁, . . . , qₖ run through the entire state space Ω‎). In order to define abstract Hilbert space, this chapter takes as a basis the fundamental vector operations af, f ± g, (f, g).

Keywords:   Hilbert space, abstract Hilbert space, mathematical basis, geometry, closed linear manifolds, eigenvalue problem, commutative operators

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