5.8 Cauchy-Schwarz Inequality

Types: N/A
Examples: N/A
Constructions: 5.4 Algebraic Properties of the 2-norm of a Vector
Generalizations: N/A

Properties: N/A
Sufficiencies: 5.7 Cosine Formula for Inner Product
Questions: N/A

Cauchy-Schwartz Inequality

Let $n \in N$ be a positive integer and let $\vec{x}, \vec{y} \in R^{n}$ . Then,

| \vec{x} \cdot \vec{y} | \leq | | \vec{x} | | | | \vec{y} | |

\begin{proof}
Let $\vec{x}, \vec{y} \in R^{n}$ .

\begin{aligned} | \vec{x} \cdot \vec{y} | & | | \vec{x} | | | | \vec{y} | | \end{aligned}

By the 5.7 Cosine Formula for Inner Product, $\vec{x} \cdot \vec{y} = | | \vec{x} | | | | \vec{y} | | \cos θ$ ,

⟹ - | | \vec{x} | | \cdot | | \vec{y} | | \leq \vec{x} \cdot \vec{y} \leq | | \vec{x} | | \cdot | | \vec{y} | |

If $- a \leq x \leq a$ , then $| x | < a$ . Therefore, the absolute value of the inner product is less than the product of two 2-norms.

\end{proof}