A treatment of the Cauchy--Schwarz inequality in $C^{\ast }$-modules

We study the Cauchy–Schwarz and some related inequalities in a semi-inner product module over a $C^{\ast }$-algebra $\text{A}$. The key idea is to consider a semi-inner product $\text{A}$-module as a semi-inner product $\text{A}$-module with respect to another semi-inner product. In this way, we improve some inequalities such as the Ostrowski inequality and an inequality related to the Gram matrix. The induced semi-inner products are also related to the the notion of covariance and variance. Furthermore, we obtain a sequence of nested inequalities that emerges from the Cauchy–Schwarz inequality. As a consequence, we derive some interesting operator-theoretical corollaries. In particular, we show that the sequence arising from our construction, when applied to a positive element of a $C^{\ast }$-algebra, converges to its pseudo-inverse.