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## Main Question or Discussion Point

In the derivation of triangle inequality [tex]|(x,y)| \leq ||x|| ||y||[/tex] one use some ##z=x-ty## where ##t## is real number. And then from ##(z,z) \geq 0## one gets quadratic inequality

[tex]||x||^2+||y||^2t^2-2tRe(x,y) \geq 0[/tex]

And from here they said that discriminant of quadratic equation

[tex]D=4(Re(x,y))^2-4 ||y||^2|x||^2 \leq 0 [/tex]

Could you explain me why ##<## sign in discriminant relation? When discriminant is less then zero solutions are complex conjugate numbers. But I still do not understand the discussed inequality. What about for example in case

[tex]||x||^2+||y||^2t^2-2tRe(x,y) \leq 0[/tex]?

[tex]||x||^2+||y||^2t^2-2tRe(x,y) \geq 0[/tex]

And from here they said that discriminant of quadratic equation

[tex]D=4(Re(x,y))^2-4 ||y||^2|x||^2 \leq 0 [/tex]

Could you explain me why ##<## sign in discriminant relation? When discriminant is less then zero solutions are complex conjugate numbers. But I still do not understand the discussed inequality. What about for example in case

[tex]||x||^2+||y||^2t^2-2tRe(x,y) \leq 0[/tex]?