## The Polarization Identities

If we have an inner product on a real or complex vector space, we get a notion of length called a “norm”. It turns out that the norm completely determines the inner product.

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.

[...] the other hand, what if we have a norm that satisfies this parallelogram law? Then we can use the polarization identities to define a unique inner [...]

Pingback by The Parallelogram Law « The Unapologetic Mathematician | April 24, 2009 |

[...] now we have the polarization identities to work with! The real and imaginary parts of are completely determined in terms of expressions [...]

Pingback by Positive-Definite Transformations « The Unapologetic Mathematician | July 13, 2009 |

[...] if for all vectors , then we can use the polarization identities to conclude that [...]

Pingback by Normal Transformations « The Unapologetic Mathematician | August 5, 2009 |

[...] at a given point, while the line element is a quadratic function of a single vector. However, the polarization identities will allow you to recover the bilinear function from the quadratic [...]

Pingback by Bad Language: Metric vs Metric Tensor vs Matrix Form vs Line Element « The Language of Bad Physics | October 21, 2009 |

[...] verify our assertion for the product , we turn and recall the polarization identities from when we worked with inner products. Remember, they told us that if we know how to calculate [...]

Pingback by Adding and Multiplying Measurable Real-Valued Functions « The Unapologetic Mathematician | May 7, 2010 |