What is the range of $\frac{\left( \sum_i x_i y_i \right)^2}{\sum_i x_i^2 \sum_i y_i^2}$ for $x, y \in \mathbb{R}^n$? i.e., is it bounded above and below by some finite number?
I know that \begin{align} \sum x_i^2 = ||x||^2 \\ \sum y_i^2 = ||y||^2 \\ \left(\sum_i(x_iy_i) \right)^2 = \langle x, y\rangle^2 \end{align}
The numerator and denominator are both positive, so the lower bound must be at least $0$. Is there a relationship between the numerator and denominator that would help me find the bounds?
Use the formula on the scalar product $$\langle x,y \rangle = |x| \cdot |y| \cos \alpha$$
You hence get that the range is $(0,1)$. Note that this is a bit circular because that is exactly how the angle between two vectors in higher dimensional space is defined (although you can embed them in a plane and get an angle like that).
Another possibility is the Cauchy-Schwatz inequality and you get an upper bound of $1$. The lower bound of $0$ is clear.