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A Hilbert space $H$ is a real or complex vector space endowed with an inner product such that $H$ is a complete metric space when endowed with the norm induced by this inner product.

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Dual operators between Hilbert spaces: with or without Riesz representation

Like Mr Yuan suggested, call the first one 'dual' and write $f^\ast$ and the second one adjoint and write $f^\dagger$. Then a fairy simple calculation shows, that $f^\ast$ and $f^\dagger$ are closely …
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