Understanding Proof of Cauchy-Schwarz Inequality The following is part of the proof for the Cauchy-Schwarz inequality from Appendix C (Linear Spaces Review) of Introduction to Laplace Transforms and Fourier Series, Second Edition, by Phil Dyke:

I'm struggling to understand the following:


*

*$- \lambda \langle \alpha \mathbf{b}, \mathbf{a} \rangle - \langle \mathbf{a}, \alpha \mathbf{b} \rangle$ to $- \lambda \alpha \langle \mathbf{b}, \mathbf{a} \rangle - \lambda \bar{\alpha}\langle \mathbf{a}, \mathbf{b} \rangle$

*$- \lambda \alpha \langle \mathbf{b}, \mathbf{a} \rangle - \lambda \bar{\alpha}\langle \mathbf{a}, \mathbf{b} \rangle$ to $-2\lambda |\alpha|^2$
I would greatly appreciate it if people could please take the time to clarify these steps.

The term-by-term steps going to the second line of the expansion are
$$
\lambda \langle\alpha b,a \rangle \to \lambda \alpha \langle b,a\rangle\\
\lambda \langle a,\alpha b \rangle \to \lambda \bar \alpha \langle a,b \rangle
$$
From there, we note that $\langle a,b \rangle = \alpha$, and $\langle b,a \rangle = \bar \alpha$.  In the last step, we observe that $\alpha \bar \alpha = |\alpha|^2$.
