1.5 Properties of inner products

Inner products and their induced norms have some very useful properties:

• Cauchy-Schwartz Inequality: $\left|⟨x,y⟩\right|\le ∥x∥∥y∥$ with equality iff $\exists \alpha \in \mathbb{C}$ such that $y=\alpha x$
• Pythagorean Theorem: $⟨x,y⟩=0\Rightarrow {∥x,+,y∥}^2={∥x,-,y∥}^2={∥x∥}^2+{∥y∥}^2$
• Parallelogram Law: ${∥x,+,y∥}^2+{∥x,-,y∥}^2=2{∥x∥}^2+2{∥y∥}^2$
• Polarization Identity: $\text{Re}\left[⟨x,y⟩\right]=\frac{{∥x,+,y∥}^2-{∥x,-,y∥}^2}{4}$

In ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$ , we are very familiar with the geometric notion of an angle between two vectors. For example, if $x,y\in {\mathbb{R}}^2$ , then from the law of cosines, $⟨x,y⟩=∥x∥∥y∥cos\theta$ . This relationship depends only on norms and inner products, so it can easily be extended to any inner product space.

Definition 1

Definition 2