Cauchy Schwarz Inequality

$|⟨\alpha |\beta ⟩|\le ||\alpha ||||\beta ||$

Specific Domains

• Cauchy Schwarz Inequality Integrals
• Cauchy Schwarz Inequality Dot Product
• Cauchy Schwarz Inequality Trace

Proof

Case 1: $\alpha =a\beta$ (They are scalars of eachother)

• Then, $|⟨\alpha |\beta ⟩|=|⟨a\beta |\beta ⟩|$
• $=|a|⟨B|B⟩$
• $=\sqrt{|a{|}^2⟨\beta |\beta {⟩}^2}$
• $=||\alpha ||\cdot ||\beta ||$

Case 2: $\alpha \ne a\beta$

• Consider $⟨\alpha =a\beta |\alpha -a\beta ⟩$
• By the fourth property of Inner Products, we know $⟨\alpha -a\beta |\alpha -a\beta ⟩>0$
• $⟹⟨\alpha |\alpha ⟩-⟨\beta |\alpha ⟩-⟨\alpha |\beta ⟩-⟨\beta |\beta ⟩$
• We choose $a=\frac{⟨\alpha |\beta ⟩}{⟨\beta |\beta ⟩}$
• Then, $⟹⟨\alpha \alpha ⟩⟨\beta |\beta ⟩-⟨\alpha |\beta ⟩⟨\beta |\alpha ⟩>0$
• $⟹|⟨\alpha |\beta ⟩|<||\alpha ||||\beta ||$