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This tag is for questions relating to Hilbert Space, a vector space equipped with an inner product, an operation that allows defining lengths and angles, and the space is complete. It arises naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces having the property that it is complete. Applies also to pre-Hilbert spaces, rigged Hilbert spaces, and spaces with negative norm or zero-norm states.

A Hilbert space, named by von Neumann after David Hilbert, is an inner product space that is complete with respect to the norm defined by the inner product.

This notion was invented by J. von Neumann in his famous book about the mathematical foundation of Quantum Theory to formalise the space of states of a quantum system.

The theory of Hilbert spaces that Hilbert, Riesz, von Neumann and others developed has not only greatly enriched the world of mathematics but has proven extremely useful in the development of scientific theories, particularly . For example, the ability to treat functions as vectors in a Hilbert space, as permitted by Hilbert space theory, has enabled quantum physicists to solve difficult differential and integral equations by using mere algebra. What is more, the theory and notation of Hilbert space has become so ingrained in the world of quantum mechanics that it is commonly used to describe many interesting phenomenon, including the EPR paradox (entanglement), quantum teleportation, and quantum telecloning.

References:
$1.~~$ "A BRIEF INTRODUCTION TO HILBERT SPACE AND QUANTUM LOGIC"
$2.~~$ "Hilbert Space Quantum Mechanics"
$3.~~$ "Hilbert Space Operators in Quantum Physics"??
$4.~~$ "Hilbert spaces"
$5.~~$ "Hilbert space"