Hilbert Space
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Hilbert space is a inner product space that is also a complete metric space. A Hilbert space is always a Banach space, but the converse need not hold. Hilbert space is named after mathematician David Hilbert, whom used it to provide a natural context in which to generalize the concept of Fourier series and Fourier Transformation in terms of arbitrary orthogonal functions defined on infinite dimensional inner product space.