theorems are further examples in operator algebra. This extends the fact when is a linear functional. The case when is self-adjoint, that is, is sometimes known as Kadison's inequality. Another generalization is a refinement obtained by interpolating between both sides of the Cauchy-Schwarz inequality: This theorem can be deduced from Hölder's inequality. There are also non commutative versions for operators and tensor products of matrices. A survey of matrix versions of Cauchy-Schwarz inequality and Kantorovich inequality is available. See also Notes Citations References . . External links Earliest Uses: The entry on the Cauchy–Schwarz inequality has some historical information. Example of application of Cauchy–Schwarz inequality to determine Linearly Independent Vectors Tutorial and Interactive program. Augustin-Louis Cauchy Inequalities Linear algebra Operator theory Articles containing proofs Mathematical analysis Probabilistic inequalities