Say I have an inner product space $(\mathbb{V}, \langle.,.\rangle)$. Now I want to prove the inequality (without using Cauchy Schwarz):
$$\|u\|^2 + \|v\|^2 \geq 2 |\langle u,v \rangle|$$.
So I started using the following property of inner products:
$$\langle v-u,v-u\rangle \geq 0$$ which will simplify to:
$$\langle v,v\rangle + \langle u,u\rangle \geq \langle v,u\rangle + \langle u,v\rangle$$
and now I am stuck. Because $\mathbb{V}$ can be a complex vector space I don't know how to move forward. How to get the required statement?
Best Answer
Let $c$ be a complex number of unit modulus such that $\langle u,cv\rangle=|\langle u,v\rangle|$. The inequality in question then follows from $\langle u-cv,u-cv\rangle\ge0$.