From the point of view of manifolds and curvature the following result is valid:

Also 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

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