## Monday Math 12: Cauchy-Schwarz

The Cauchy-Schwarz Inequality

A simple, but important theorem in linear algebra is the Cauchy-Schwarz inequality, which applies to all vector spaces with an inner product. We have the norm as usually defined on an inner product space: .
We will examine the proof for a complex vector space (the proof for a real vector space is analogous).

Assume we have vectors v and w, w0; and scalar c. Then we know:






Now, let , which is valid as w is nonzero.
Then , and our above inequality becomes:




Since the terms being squared on both sides are positive, we can take the square root of both sides to obtain:

This is the Cauchy-Schwarz inequality. Note that it also holds (trivially) for w=0. It can also be shown that equality holds only when the two vectors are linearly dependent.