Given two vectors $\mathbf{u}$ and $\mathbf{v}$ in an [inner product space](Inner%20products.md#Inner%20product%20spaces) where the [vector normalization](Vector%20normalization.md) is defined as $||u||=\sqrt{(\mathbf{u}, \mathbf{u})},$ the _Cauchy-Schwarz inequality_ (occasionally referred to as the _Schwarz inequality_) is as follows: $|(\mathbf{u}, \mathbf{v})| \leq ||\mathbf{u}||\,||\mathbf{v}||$ The Cauchy-Schwarz inequality is a specific case of [[Hölder's inequality]]. %%This needs to also be elaborated on for normed function spaces%% --- # Proofs and examples ## Proof of the Cauchy-Schwarz inequality on [complex vector space](Complex%20vector%20spaces.md)s        ## General proof of the Cauchy-Schwarz inequality --- # Recommended reading The [Cauchy-Schwarz inequality](Cauchy-Schwarz%20inequality.md) for [complex vector spaces](Complex%20vector%20spaces.md) is a relevant property for vectors in quantum mechanics, this is described in the following mathematical physics and quantum physics textbooks: * [von Neumann J., _Mathematical Foundations of Quantum Mechanics_. Translated by Robert T. Beyer. Princeton University Press, 2018.](von%20Neumann%20J.,%20Mathematical%20Foundations%20of%20Quantum%20Mechanics.%20Translated%20by%20Robert%20T.%20Beyer.%20Princeton%20University%20Press,%202018..md) pgs 28-29. The [proof of the Cauchy-Schwarz inequality on complex vector spaces](Cauchy-Schwarz%20inequality.md#Proof%20of%20the%20Cauchy-Schwarz%20inequality%20on%20complex%20vector%20space%20Complex%2020vector%2020spaces%20md%20s) follows closely from this section of the text. %%Find your proof of the Schwarz inequality from page 34 of Sakurai's quantum mechanics %% %%It is more than just for quantum mechanics. It is a key theorem for analysis and functional analysis%% #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra #MathematicalFoundations/Analysis/FunctionalAnalysis