Let $\mathcal{X}$ be a real or complex Banach space.
It is a well known fact that $\mathcal{X}$ is a Hilbert space (i.e. the norm comes from an inner product) if the parallelogram identity holds.


Question: Are there other (simple) characterizations for a Banach space to be a Hilbert space?



When is a Banach space a Hilbert space?
From [this article](https://doi.org/10.3103/S0027132208050070) by O. N. Kosukhin:



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> A real Banach space $(X, \|\cdot\|)$ is a Hilbert space if and only if for any three points $A$, $B$, $C$ of this space not belonging to a line there are three altitudes in the triangle $ABC$ intersecting at one point.
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Many other references show when Googling



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> "is a hilbert space if" banach
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