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Question: Given two Banach spaces $X$ and $Y$, if there exists an onto linear isometry $T:X\to Y$ and $X$ is a Hilbert space, is it true that $Y$ is a Hilbert space?

Intuitively this seems true to me. To prove this, one just need to construct an $\langle\cdot, \cdot \rangle$ on $Y\times Y$ and show that it is an inner product.

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