Let $x_1, y_1, \ldots, x_N, y_N \in [0, \infty)$ each be an D4767: Unsigned real number.

Then

(1)	\begin{equation} \sum_{n = 1}^N x_n y_n \leq \left( \sum_{n = 1}^N x_n^2 \right)^{1 / 2} \left( \sum_{n = 1}^N y_n^2 \right)^{1 / 2} \end{equation}
(2)	\begin{equation} \sum_{n = 1}^N x_n y_n = \left( \sum_{n = 1}^N x_n^2 \right)^{1 / 2} \left( \sum_{n = 1}^N y_n^2 \right)^{1 / 2} \quad \iff \quad \exists \, c \in [0, \infty) : x = c y \text{ or } y = c x \end{equation}