Here is an homework that revisits  (in the simplest case of real vector space) the proof of 1935 by Jordan and von Neumann, that a vector normed space satisfying the parallelogram law can be equiped with an inner product: Borboleta_2012_Homework-on-Jordan-Neumann-Inner-Product-Spaces-Characterization.pdf.

 In this proof, the triangular inequality proprety of the norm is invoked for a continuity argument, but otherwise not used.