TITLE: When is a Banach space a Hilbert space?
QUESTION [32 upvotes]: 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?

REPLY [7 votes]: From the point of view of manifolds and curvature the following result is valid:
A Banach space is a Hilbert space if and only if it is a NPC (non-positive curvature) space.
 http://www.iam.uni-bonn.de/fileadmin/WT/Inhalt/people/Karl-Theodor_Sturm/papers/paper41.pdf