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[tex]a = \lambda b[/tex],

then the Cauchy-Schwartz inequality

[tex]

\newcommand{\braket}[2]{{<\!\!{#1|#2}\!\!>}}

|\braket{a}{b}|^2 \leq \braket{a}{a} \braket{b}{b}

[/tex]

is an equality.

But how do I show that it is an equality if and only if they are parallel?