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I was diving into my 3rd year quantum assignment and I saw the following which I have to use for the rest of the question to prove the Cauchy-Schwarz inequality:

## Homework Statement

[tex] || a|x> + b|y> ||^2 [/tex]

I only really learned a bit about Dirac notation last year, so please let me know if this simplification is right.

[tex] ( a|x> + b|y> )^{*}( a|x> + b|y> ) [/tex]

[tex] a^{*}a<x|x> + a^{*}b<x|y> + b^{*}a<y|x> + b^{*}b<y|y> [/tex]

Later it says to make substitutions for [tex] a= -<x|y> , b= <x|x> [/tex] to prove the inequality, but I want to make sure I didn't screw up the very first part above...which I have a feeling I did.

Thanks