While reading the proof of Cauchy-schwarz inequality, I didn't get one step.
The step is as below,
by positivity axiom, for any real number $t$
$0≤⟨tu+v,tu+v⟩=⟨u,u⟩t^2+ 2⟨u,v⟩t+⟨v,v⟩$
This imply $$0≤at^2 + bt+c$$ where $a=⟨u,u⟩$, $b=2⟨u,v⟩$ and $c=⟨v,v⟩$
After this they had written, this inequality implies that the quadratic polynomial has either no real roots or repeated real roots!
I didn't get this! How the quadratic polynomial $at^2 + bt+c$ has either no real roots or repeated real root?
Best Answer
You can even think in a geometric way. If $at^2+bt+c\geq 0$ for all $t\in\mathbb{R}$ then the parabola is never below the real axis. What does it tell us about the number of times it intersects the real axis?