Inner products are bounded by the product of norms, with equality exactly when the two vectors are linearly dependent.Suggest a slogan
Let \(H\) be an inner product space over \(\CC\) with inner product \(\inner{\cdot}{\cdot}\) and norm \(\norm{x}\coloneqq\inner{x}{x}^{1/2}\). For all \(f,g\in H\), \[ \abs{\inner{f}{g}} \leq \norm{f}\norm{g}, \] with equality if and only if \(f\) and \(g\) are linearly dependent. In particular, for a measure space \((X,\mcm,\mu)\) and \(f,g\in L^2(\mu)\), \(\abs{\int_X f\overline g\dmu}\leq\norm{f}_2\norm{g}_2\).