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INTRODUCTION Hilbert spaces were named after David Hilbert , who studied them in the context of integral equations. John Von Neumann originated the designation "der abstrakte Hilbertsche Raum" in his famous work on unbounded Hermitian Operators published in 1929 . Von Neumann was perhaps the mathematician who most clearly recognized their importance as a result of his seminal work on the foundations of quantum mechanics begun with Hilbert and Lothar (Wolfgang) Nordheim and continued with Eugene Wigner . The name "Hilbert space" was soon adopted by others, for example by Hermann Weyl in his book ''The Theory of Groups and Quantum Mechanics'' published in 1931 (English language paperback ISBN 0486602699). The elements of an abstract Hilbert space are sometimes called "vectors". In applications, they are typically Sequence s of Complex Number s or Function s. In quantum mechanics for example, a physical system is described by a complex Hilbert space which contains the " Wavefunction s" that stand for the possible states of the system. See Mathematical Formulation Of Quantum Mechanics for details. The Hilbert space of Plane Wave s and Bound State s commonly used in quantum mechanics is known more formally as the Rigged Hilbert Space . DEFINITION | ||
| :<math>\x\ | \sqrt{\langle x,x
angle}</math> |
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| :<math> \ell^2(B) | \left\{ x:B
ightarrow \mathbb{C}\,\bigg\,\sum_{b \in B} \leftx \left(b
ight)
ight^2 < \infty
ight\}</math> |
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| # Elements Are Normalized: Every Element Of The Family Has Norm 1: ''e''<sub>''k''</sub> | 1 for all ''k'' in ''B'' |
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| '''Theorem''' The Orthogonal Projection P<sub>''V''</sub> Is A Self-adjoint Linear Operator On ''H'' Of Norm &le 1 With The Property P<sub>''V''</sub><sup>2</sup> | P<sub>''V''</sub> Moreover, any self-adjoint linear operator ''E'' such that ''E''<sup>2</sup> = ''E'' is of the form P<sub>''V''</sub>, where ''V'' is the range of ''E'' For every ''x'' in ''H'', P<sub>''V''</sub>(''x'') is the unique element ''v'' of ''V'' which minimizes the distance ''x'' - ''v'' |