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An isomorphism of Hilbert spaces, $H \xrightarrow{\varphi} G$, is a linear mapping that is one-to-one and onto, and with $\left(\varphi(h), \varphi\left(h^{\prime}\right)\right)=\left(h, h^{\prime}\right)$. Prove that the free Hilbert spaces on sets $K$ and $K^{\prime}$ are isomorphic if and only if these sets are isomorphic.

   An isomorphism of Hilbert spaces, $H \xrightarrow{\varphi} G$, is a linear mapping that is one-to-one and onto, and with $\left(\varphi(h), \varphi\left(h^{\prime}\right)\right)=\left(h, h^{\prime}\right)$. Prove that the free Hilbert spaces on sets $K$ and $K^{\prime}$ are isomorphic if and only if these sets are isomorphic.

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Mathematical physics

Mathematical physics

Robert Geroch 1st Edition

Chapter 48, Problem 332

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We need to show that the free Hilbert spaces on sets \( K \) and \( K' \) are isomorphic if and only if the sets \( K \) and \( K' \) themselves are isomorphic. A free Hilbert space on a set \( K \) is essentially the space of square-summable sequences indexed by Show more…

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An isomorphism of Hilbert spaces, $H \xrightarrow{\varphi} G$, is a linear mapping that is one-to-one and onto, and with $\left(\varphi(h), \varphi\left(h^{\prime}\right)\right)=\left(h, h^{\prime}\right)$. Prove that the free Hilbert spaces on sets $K$ and $K^{\prime}$ are isomorphic if and only if these sets are isomorphic.

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