ThmDex – An index of mathematical definitions, results, and conjectures.

Cauchy-Schwarz inequality for two real sequences

Formulation 0

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}

Formulation 1

Let $x, y \in [0, \infty)^{N \times 1}$ each be an D6043: Unsigned real column matrix.

Then

(1)	\begin{equation} x^T y \leq \Vert x \Vert_2 \Vert y \Vert_2 \end{equation}
(2)	\begin{equation} x^T y = \Vert x \Vert_2 \Vert y \Vert_2 \quad \iff \quad x \text{ and } y \text{ are linearly dependent} \end{equation}

Proofs

Proof 0

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

This result is a particular case of R5190: Weighted Cauchy-Schwarz inequality for two real sequences. $\square$