Note: Due to the nature of HTML publishing the inner product notation will be square brackets instead of the pointed brackets.
Let (V,|| ||) be a normed vector space.
Then || || comes from an inner product <=> the norm satisfies the parallelogram law
|| u + v ||^2 + || u-v||^2 = 2(||u||^2 + ||v||^2)
for all u,v in V.
V is a real vector space with a function
|| ||:V->R satisfying
If || || comes from an inner product then:
[u+v,u+v] + [u-v,u-v] using the definition of a norm
[u,u]+[v,v]+[u,v]+[u,v]+[u,u]+[v,v]-[v,u]-[u,v] using bilinearity of [,] and symmetry.
2([u,u]+[v,v]) by using distributive property of R
2(||u||^2 + ||v||^2) by definition of a norm of a vector space