In my lecture notes I've written the proof of Cauchy-Schwarz inequality as:

Let t $\in$ R and $\langle x+ty, x+ty\rangle \geq 0$, then

$\langle x+ty, x+ty\rangle $ = $\langle x, x+ty \rangle + t\langle y, x+ty\rangle$

= $\langle x, x \rangle + t \langle x, y \rangle + t \langle y, x \rangle + t^{2}\langle y, y \rangle$

= $\| x \|^{2} +2t \langle x, y \rangle + t^{2} \| y\|^{2} \geq 0$

On the second line of the proof, I wonder if I forgot to write a $t$ before $\langle x, x+ty \rangle $, and also what is $t$'s purpose in the proof?

  • $\begingroup$ There are two things that I want to mention. By writing $\langle x + ty, x + ty \rangle \geq 0$ you're comparing a vector to a number. The first equality you have should read $\langle x+ty, x+ty\rangle $ = $\langle x, x+ty \rangle + t\langle y, 0\rangle$ . Among other things, this approach to proving the inequality might not be best. $\endgroup$ – ThisIsNotAnId Feb 18 '16 at 7:05
  • $\begingroup$ The proof given for this on the corresponding Wikipedia page might help en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality $\endgroup$ – ThisIsNotAnId Feb 18 '16 at 7:16

You didn't forget a $t$ at the second line before $\langle x,x+ty\rangle$ and you have to use the fact that it is a polynomial of degree 2 in $t$ so the discriminant is $\leq 0$.


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