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I have a question in a Hilbert Spaces course as follows: Let $X=(x_1, x_2)$ be vector in a vector space of all ordered pairs of complex numbers X. Can we obtain the norm defined on X by: $\|X\|=|x_1|+|x_2|$ from an inner product?

I know a norm comes from an inner product space if and only if it satisfies the parallelogram law given by $\|X+Y\|^2+\|X-Y\|^2=2\|X\|^2+2\|Y\|^2$. However, I am quite stuck using this method. Is there another way to solve it?

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No, you can't obtain this norm from an inner product. For instance, the parallelogram law doesn't hold for $X=(1,0)$ and $Y=(0,1)$:

$$ \|X+Y\|^2+\|X-Y\|^2=(|1|+|1|)^2+(|1|+|-\!1|)^2=8\ne2\|X\|^2+2\|Y\|^2\;. $$

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    $\begingroup$ Thank you! that was very helpful. $\endgroup$ – user120768 May 15 '16 at 13:08

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