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Hilbert space

A Hilbert space is a vector space with an inner product such that the norm defined byturns into a complete metric space. If the metric defined by the norm is not complete, then is instead known as an inner product space.Examples of finite-dimensional Hilbert spaces include1. The real numbers with the vector dot product of and . 2. The complex numbers with the vector dot product of and the complex conjugate of . An example of an infinite-dimensional Hilbert space is , the set of all functions such that the integral of over the whole real line is finite. In this case, the inner product isA Hilbert space is always a Banach space, but theconverse need not hold.A (small) joke told in the hallways of MIT ran, "Do you know Hilbert? No? Then what are you doing in his space?" (S. A. Vaughn, pers. comm., Jul. 31, 2005)...

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