Cauchy-Schwarz inequality

Let V be a vector spaceMathworldPlanetmath where an inner productMathworldPlanetmath , has been defined. Such spaces can be given also a norm by defining


Then in such a space the Cauchy-Schwarz inequality holds:


for any v,wV. That is, the modulus (since it might as well be a complex numberMathworldPlanetmathPlanetmath) of the inner product for two given vectors is less or equal than the product of their norms. Equality happens if and only if the two vectors are linearly dependent.

A very special case is when V=n and the inner product is the dot productMathworldPlanetmath defined as v,w=vtw and usually denoted as vw and the resulting norm is the Euclidean normMathworldPlanetmath.

If 𝐱=(x1,x2,,xn) and 𝐲=(y1,y2,,yn) the Cauchy-Schwarz inequality becomes


which implies


Notice that in this case inequalityMathworldPlanetmath holds even if the modulus on the middle term (which is a real number) is not used.

Cauchy-Schwarz inequality is also a special case of Hölder inequality. The inequality arises in lot of fields, so it is known under several other names as Bunyakovsky inequality or Kantorovich inequality. Another form that arises often is Cauchy-Schwartz inequality but this is a misspelling since the inequality is named after history/Mathematicians/Schwarz.htmlHermann Amandus Schwarz (1843–1921).

This inequality is similar to the triangle inequalityMathworldMathworldPlanetmath, talking about products instead of sums:

𝐱+𝐲𝐱+𝐲   triangle inequality
|𝐱𝐲|𝐱𝐲   CS inequality
Title Cauchy-Schwarz inequality
Canonical name CauchySchwarzInequality
Date of creation 2013-03-22 12:14:46
Last modified on 2013-03-22 12:14:46
Owner drini (3)
Last modified by drini (3)
Numerical id 20
Author drini (3)
Entry type Theorem
Classification msc 15A63
Synonym Kantorovich inequality
Synonym Bunyakovsky inequality
Synonym Schwarz inequality
Synonym Cauchy inequality
Synonym CBS inequality
Related topic InnerProduct
Related topic InnerProductSpace
Related topic DotProduct
Related topic NormedVectorSpace
Related topic VectorNorm
Related topic CauchySchwartzInequality
Related topic HolderInequality
Related topic CauchySchwarzInequality
Related topic VectorPnorm
Defines Cauchy-Schwartz inequality