inner product

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 ${{\\bf v}}_{1},{{\\bf v}}_{2},{{\\bf v}},{{\\bf w}}\\in V$ , the following properties hold:

1.
$(k_{1}{{\\bf v}}_{1}+k_{2}{{\\bf v}}_{2},{{\\bf w}})=k_{1}({{\\bf v}}_{1},{{\\bf w}%})+k_{2}({{\\bf v}}_{2},{{\\bf w}})$ (linearity¹¹A small minority of authors impose linearity on the second coordinate instead of the first coordinate.)
2.
$({{\\bf v}},{{\\bf w}})=\\overline{({{\\bf w}},{{\\bf v}})}$ , where $\\overline{\\ \\ \\ \\ }$ denotes complex conjugation (conjugate symmetry)
3.
$({{\\bf v}},{{\\bf v}})\\geq 0$ , and $({{\\bf v}},{{\\bf v}})=0$ if and only if ${{\\bf v}}={{\\bf 0}}$ (positive definite)

(Note: Rule 2 guarantees that $({{\\bf v}},{{\\bf v}})\\in\\mathbb{R}$ , so the inequality $({{\\bf v}},{{\\bf v}})\\geq 0$ 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 $||{{\\bf v}}||:=\\sqrt{({{\\bf v}},{{\\bf v}})}$ .