**Is there a more physical description for the Schwarz Inequality? It almost looks like it could be applied to QM expectation values but it lacks the matrix operator we expect so I’m not quite sure if it’s applicable.**

\left | \left < a | b \right> \right |^2 \le \left < a | a \right> \left < b | b \right>

The Schwarz inequality is a restriction on the magnitude of the dot product between two vectors and the lengths of the two vectors. For spatial vectors, the Schwarz inequality is essentially a statement that \cos(\theta) is less than or equal to 1 where \theta in the angle between the two spatial vectors. The same concept holds in a generalized vector space, just in a more abstract way.