Edit : misread question.
I think it's true: If the underlying field of is then (because ) and (where this last means the norm induced by the scalar product) doing the same for , by the Cauchy-Schwarz inequality we have with equality iff are l.d. which proves the statement. The real case is easier
Thanks Jose27. Yes - I get your point when you say it is a form of Cauchy-Schwarz inequality. But I would appreciate if someone can help me prove that
"the equality holds IFF u,v are linearly dependent"
So basically I want to start from <u, v> <v,u> = <u,u> <v,v> and prove u,v and lineraly dependent