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I have to ask a dumb question. I seem to be doing something very wrong here, and it's probably trivial, but for some reason I don't see what it is. I decided to try to prove the Cauchy-Schwarz inequality without opening a book. I remember that a proof I read once started by noting that
0\leq(x+ty,x+ty)=||x||^2+2 Re (t(x,y))+||y||^2
t is an arbitrary complex number. From here on it's supposed to be about choosing the right t. The inequality above is obviously satisfied when Re(t(x,y))>0, so let's choose Arg t so that it's negative instead.
We get
0\leq||x||^2-2|t||(x,y)|+||y||^2
or equivalently
|(x,y)|\leq\frac{1}{2|t|}(||x||^2+||y||^2)
Now if we choose
|t|=\frac{||x||^2+||y||^2}{2||x||||y||}
the inequality above turns into the Cauchy-Schwarz inequality.
Here's my dumb question: By choosing |t| much larger, I can make |(x,y)| as small as I want, and this implies that (x,y)=0 for arbitrary x and y! This obviously can't be true, I must have done something wrong. What's my mistake?
By the way, is there a better way to write the norm in LaTeX?
0\leq(x+ty,x+ty)=||x||^2+2 Re (t(x,y))+||y||^2
t is an arbitrary complex number. From here on it's supposed to be about choosing the right t. The inequality above is obviously satisfied when Re(t(x,y))>0, so let's choose Arg t so that it's negative instead.
We get
0\leq||x||^2-2|t||(x,y)|+||y||^2
or equivalently
|(x,y)|\leq\frac{1}{2|t|}(||x||^2+||y||^2)
Now if we choose
|t|=\frac{||x||^2+||y||^2}{2||x||||y||}
the inequality above turns into the Cauchy-Schwarz inequality.
Here's my dumb question: By choosing |t| much larger, I can make |(x,y)| as small as I want, and this implies that (x,y)=0 for arbitrary x and y! This obviously can't be true, I must have done something wrong. What's my mistake?
By the way, is there a better way to write the norm in LaTeX?
