Real Inner Product

Definition

The real inner product (or dot product, scalar product) of two real vectors $x, y \in \R ^n$ is

\[ x_1y_1 + x_2y_2 + \cdots + x_ny_n \]

Properties

The inner product has several important properties

1. $\ip{\alpha x, y} = \alpha\ip{x, y}$
2. $\ip{x + y, z} = \ip{x, z} + \ip{y, z}$
3. $\ip{x, y} = \ip{y, x}$
4. $\ip{x,x} \geq 0$
5. $\ip{x,x} = 0 \iff x = 0$

Connection to norm

It is important to note that $\norm{x} = \sqrt{\ip{x,x}}$.