I have a question regarding problem 19 in the 3rd Ed. of Spivak's Calculus. Specifically, part (a). The question concerns the Schwarz inequality:
$$
x_1y_1 + x_2y_2 \leq \sqrt{x_1^2+x_2^2}\sqrt{y_1^2+y_2^2} \ .
$$
It says to prove that if $x_1=\lambda y_1$ and $x_2 = \lambda y_2$ for some number $\lambda$, then equality holds in the Schwarz inequality.
Substituting the given values for $x_1$ and $x_2$ we have
$$
\lambda (y_1^2+ y_2^2) \leq |\lambda|(y_1^2+y_2^2) \ .
$$
It appears to me that equality can only hold if $\lambda \geq 0$. Can someone explain to me how equality holds for any given $\lambda$?
Best Answer
That is a typo. You need $\lambda\ge 0$.