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Also known as the Riesz-Fréchet Theorem, this is a surprising result in functional analysis that was proved by Frigyes Riesz in 1907 and independently at about the same time by Maurice René Fréchet. It basically shows that all of the bounded continuous linear functionals (but then again, there's also a theorem that says that every linear functional is bounded if and only if it is continuous) over a Hilbert space can be represented as an inner product, i.e., for every bounded linear functional F in a Hilbert space H, there exists a yF in H such that:

F x = <x|yF>

This theorem basically shows that a Hilbert space and its dual space are isometrically isomorphic (anti-isomorphic if the base field is the field of complex numbers). It is also the justification for the use of the bra-ket notation in the Dirac Formalism of Quantum Mechanics. It also provides the foundation for the reproducing kernel Hilbert spaces used in the theory of wavelets.

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