An inner product on a vector space $V$ over a field $K$ (which must be either the field $\mathbb{R}$ of real numbers or the field $\mathbb{C}$ of complex numbers) is a function $(\ ,\ ): V \times V \longrightarrow K$ such that, for all $k_1,k_2 \in K$ and $\v_1, \v_2, \v, \w \in V$ , the following properties hold:

1. $(k_1 \v_1 + k_2 \v_2, \w) = k_1 (\v_1, \w) + k_2 (\v_2, \w)$ (linearity ¹)
2. $(\v, \w) = \overline{(\w, \v)}$ , where $\overline{\ \ \ \ }$ denotes complex conjugation (conjugate symmetry)
3. $(\v, \v) \geq 0$ , and $(\v, \v) = 0$ if and only if $\v = \0$ (positive definite)

(Note: Rule 2 guarantees that $(\v,\v) \in \mathbb{R}$ , so the inequality $(\v,\v) \geq 0$ in rule 3 makes sense even when $K=\mathbb{C}$ .)

The standard example of an inner product is the dot product on $K^n$ : $$ ((x_1,\dots,x_n), (y_1,\dots,y_n)) := \sum_{i=1}^n x_i \overline{y_i} $$

Every inner product space is a normed vector space, with the norm being defined by $||\v|| := \sqrt{(\v,\v)}$ .

Footnotes

... (linearity¹

A small minority of authors impose linearity on the second coordinate instead of the first coordinate.