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The Cauchy-Schwarz inequality states $|\langle x,y \rangle |\leq ||x||\cdot ||y||.$ Use this tag for questions related to the CS inequality and its applications.
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Cauchy-Schwarz inequality for bilinear forms valued in an abstract vector space
I'm not sure if this is the direction you are after, but there is a beautiful generalisation of the Cauchy-Schwarz inequality to C*-algebras, called the Kadison-Schwarz inequality. Instead of vector s …