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My mathematical methods for theoretical physics course recently began looking at linear vector spaces. We defined the Banach and Hilbert Spaces and proved the Cauchy-Shwarz Inequality. There's one step in this proof that I can't really follow (in red):
consider: w=x+uy (i'll drop the emboldening of vectors now to save time)
0 <= <x,x> + <uy,x> + <x,uy> + <uy,uy>
0 <= <x,x> + u*<y,x> + u<x,y> + u*u<y,y>
Choose u = <y,x>/<y,y>
0 <= <x,x> + (<y,x>/<y,y>)*<y,x> + (<y,x>/<y,y>)<x,y> + (<y,x>/<y,y>)2<y,y>
some manipulation here concludes the proof.
My question is that by choosing a specific u are we not losing generality?
consider: w=x+uy (i'll drop the emboldening of vectors now to save time)
0 <= <x,x> + <uy,x> + <x,uy> + <uy,uy>
0 <= <x,x> + u*<y,x> + u<x,y> + u*u<y,y>
Choose u = <y,x>/<y,y>
0 <= <x,x> + (<y,x>/<y,y>)*<y,x> + (<y,x>/<y,y>)<x,y> + (<y,x>/<y,y>)2<y,y>
some manipulation here concludes the proof.
My question is that by choosing a specific u are we not losing generality?