Cauchy-Schwarz inequality

$\Vert ⟨v,w⟩\Vert \le \Vert v\Vert \Vert w\Vert$, where

• $⟨\cdot ,\cdot ⟩$ is an Inner Product
• $v,w$ are Vectors
• $\Vert v\Vert :=\sqrt{⟨v,v⟩}$ is the Norm induced by the Inner Product

Proof

• Write out this proof. In the meantime, see the Wikipedia proofs