Q1: Let $V$ be a vector space with inner product $\langle \cdot, \cdot \rangle$. What is not a correct formulation for the Cauchy-Schwarz inequality?

A1: $\langle x, y \rangle \leq \langle x, x \rangle \langle y, y \rangle $

A2: $|\langle x, y \rangle|^2 \leq \langle x, x \rangle \langle y, y \rangle $

A3: $|\langle x, y \rangle| \leq | x | | y| $

A4: $\frac{\langle x, y \rangle}{| x | | y|} \in [-1,1] $ for $x \neq 0 \neq y$.

Q2: Let $V = \mathbb{C}$ be given with the standard inner product $\langle y, x \rangle = \overline{y} x$. What does the Cauchy-Schwarz inequality say in this case?

A1: $ | x y | \leq |x| |y| $

A2: $ | x y | < |x| |y| $

A3: $ | x y | > |x| |y| $

A4: $ | x y | \neq |x| |y| $