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Let $$\Bbb C^2=\{w=(z_1,z_2) : z_1,z_2\in\Bbb C\}$$ be the vector space of all ordered pairs of complex numbers.
Can we obtain the norm defined on $\Bbb C^2$ by $$||w||=|z_1|+|z_2|$$ from an inner product? Justify.

Is it enough to show that it satisfies the parallelogram law?

Please explain in detail. Thank you.

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    $\begingroup$ Yes, it's enough to check whether the parallelogram law holds. Look for the polarization identity. $\endgroup$ – Omnomnomnom Jul 1 '15 at 6:35

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