Theorem (Cauchy-Schwarz inequality): For \(\x, \y \in \real^d\),
\[\x \Tr \y \le \norm{\x}_2\norm{\y}_2,\]where equality holds only if \(\x=a\y\) for some scalar \(a\).
Proof: The Cauchy-Schwarz Master Class (2004), Steele JM. Cambridge University Press.
It is also common to encounter the Cauchy-Schwarz theorem in the following form (squaring both sides):
\[\left( \sum_i x_i y_i \right)^2 \le \sum_i x_i^2 \sum_i y_i^2.\]If \(\x\) and \(\y\) contain both positive and negative values, it is worth noting that the Cauchy-Schwarz theorem also applies to \(\x^*\) and \(\y^*\), where the elements of \(\x^*\) are the absolute value of the elements of \(\x\) and so on for \(\y^*\). This results in yet another form of the Cauchy-Schwarz theorem:
\[\sum_i \abs{x_i y_i} \le \norm{\x}_2 \norm{\y}_2,\]or
\[\tfrac{1}{n} \sum_i \abs{x_i y_i} \le \sqrt{\tfrac{1}{n}\sum_i x_i^2} \sqrt{\tfrac{1}{n}\sum_i y_i^2}.\]