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I have to define inner product of two spaces $(X_i, \langle \cdot, \cdot \rangle_i)$ for $i=1,2$. But there is a question, when $X_1 \times X_2$ is a Hilbert space. Cartesian product of two Hilbert spaces equipped with the inner product $\langle\cdot, \cdot\rangle_1 + \langle\cdot, \cdot\rangle_2$ is again Hilbert space. Is it possible to weaken the assumption of two spaces to be Hilbert?

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If you want $X_1\times X_2$ to be Hilbert, no: both need to be Hilbert. Consider a non-convergent Cauchy sequence $\{x^n\}_{n\in\Bbb N}$ in, say, $X_1$. Then $\{(x^n,0)\}_{n\in\Bbb N}$ is Cauchy but not convergent.

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