inner product (original) (raw)

An inner product on a vector space V over a field K (which must be either the field ℝ of real numbers or the field ℂ of complex numbers) is a function (,):V×V⟶K such that, for all k1,k2∈K and 𝐯1,𝐯2,𝐯,𝐰∈V, the following properties hold:

1. 1. (k1⁢𝐯1+k2⁢𝐯2,𝐰)=k1⁢(𝐯1,𝐰)+k2⁢(𝐯2,𝐰) (linearity11A small minority of authors impose linearity on the second coordinate instead of the first coordinate.)

(Note: Rule 2 guarantees that (𝐯,𝐯)∈ℝ, so the inequality (𝐯,𝐯)≥0 in rule 3 makes sense even when K=ℂ.)

The standard example of an inner product is the dot product on Kn: