The Polarization Identities
Let’s take the sum of two vectors and . We can calculate its norm-squared as usual:
where denotes the real part of the complex number . If is already a real number, it does nothing.
So we can rewrite this equation as
If we’re working over a real vector space, this is the inner product itself. Over a complex vector space, this only gives us the real part of the inner product. But all is not lost! We can also work out
where denotes the imaginary part of the complex number . The last equality holds because
so we can write
We can also write these identities out in a couple other ways. If we started with , we could find the identities
Or we could combine both forms above to write
In all these ways we see that not only does an inner product on a real or complex vector space give us a norm, but the resulting norm completely determines the inner product. Different inner products necessarily give rise to different norms.