inner product

An inner productMathworldPlanetmath on a vector spaceMathworldPlanetmath V over a field K (which must be either the field of real numbers or the field of complex numbersMathworldPlanetmathPlanetmath) is a function (,):V×VK such that, for all k1,k2K and 𝐯1,𝐯2,𝐯,𝐰V, the following properties hold:

  1. 1.

    (k1𝐯1+k2𝐯2,𝐰)=k1(𝐯1,𝐰)+k2(𝐯2,𝐰) (linearity11A small minority of authors impose linearity on the second coordinatePlanetmathPlanetmath instead of the first coordinate.)

  2. 2.

    (𝐯,𝐰)=(𝐰,𝐯)¯, where ¯ denotes complex conjugation (conjugatePlanetmathPlanetmathPlanetmath symmetryPlanetmathPlanetmath)

  3. 3.

    (𝐯,𝐯)0, and (𝐯,𝐯)=0 if and only if 𝐯=𝟎 (positive definitePlanetmathPlanetmath)

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

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


Every inner product spaceMathworldPlanetmath is a normed vector spacePlanetmathPlanetmath, with the norm being defined by ||𝐯||:=(𝐯,𝐯).

Title inner product
Canonical name InnerProduct
Date of creation 2013-03-22 12:13:39
Last modified on 2013-03-22 12:13:39
Owner djao (24)
Last modified by djao (24)
Numerical id 15
Author djao (24)
Entry type Definition
Classification msc 11E39
Classification msc 15A63
Synonym Hermitian inner product
Related topic InnerProductSpace
Related topic HermitianForm
Related topic EuclideanVectorSpace