Hilbert Spaces Inner product spaces Example Prove that in a real vector space with inner product we have (x, y) = 1 4 x + y 2 − x − y 2 , and in a complex vector space with inner product we have (x, y) = 1 4 x + y 2 − x − y 2 + i x + iy 2 + i x − iy 2 . These are the so-called polarization identities. They tell us that in a Hilbert space, the inner product is determined by the norm.