If VVV is an inner product space, the Cauchy-Schwarz inequality states:
∣⟨u,v⟩∣≤∥u∥∥v∥|\langle u, v \rangle| \leq \|u\| \|v\|∣⟨u,v⟩∣≤∥u∥∥v∥
⟨u,v⟩≤∥u∥+∥v∥\langle u, v \rangle \leq \|u\| + \|v\|⟨u,v⟩≤∥u∥+∥v∥
The equality holds if and only if uuu and vvv are linearly dependent
∣⟨u,v⟩∣≥∥u∥∥v∥|\langle u, v \rangle| \geq \|u\| \|v\|∣⟨u,v⟩∣≥∥u∥∥v∥