Inequalities for inner products

Tables

Equation	Form	Notes
Cauchy-Schwarz inequality	$$\vert\langle x,y\rangle\vert\le\Vert x\Vert\Vert y\Vert$$ for $\Vert x\Vert:=\sqrt{\langle x,x\rangle}$ (equality if lin dependent)	For any Inner product, note that $\Vert\cdot\Vert$ is the norm induced by the inner product
Parallelogram law	For any i.p.s we have $$\Vert x+y\Vert^2+\Vert x-y\Vert^2=2\Vert x\Vert^2+2\Vert y\Vert^2$$	Page 11
Polarisation identities	For a $\mathbb{R}$ inner product: $$\langle x,y\rangle=\frac{1}{4}\Vert x+y\Vert^2-\frac{1}{4}\Vert x-y\Vert^2$$	Page 10
	For a $\mathbb{C}$ inner product: $$\langle x,y\rangle=\frac{1}{4}\Vert x+y\Vert^2-\frac{1}{4}\Vert x-y\Vert^2+j\left[\frac{1}{4}\Vert x+jy\Vert^2-\frac{1}{4}\Vert x-jy\Vert^2\right]$$
Pythagorean theorem	If $x$ is perpendicular to $y$ in any i.p.s, then: $$\Vert x+y\Vert^2=\Vert x\Vert^2+\Vert y\Vert^2$$	Page 11