Cauchy-Schwarz inequality

Let $V$ be a vector space where an inner product $⟨,⟩$ has been defined. Such spaces can be given also a norm by defining

$$\parallel x\parallel =\sqrt{⟨x,x⟩}.$$

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

$$|⟨v,w⟩|\le \parallel v\parallel \parallel w\parallel$$

for any $v,w\in V$. That is, the modulus (since it might as well be a complex number) 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 $V={ℝ}^n$ and the inner product is the dot product defined as $⟨v,w⟩=v^{t}w$ and usually denoted as $v\cdot w$ and the resulting norm is the Euclidean norm.

If $𝐱=(x_1,x_2,\mathrm{\dots },x_n)$ and $𝐲=(y_1,y_2,\mathrm{\dots },y_n)$ the Cauchy-Schwarz inequality becomes

$$|𝐱\cdot 𝐲|=|x_1{y}_1+x_2{y}_2+\mathrm{\cdots }+x_n{y}_n|\le \sqrt{x_1^2+x_2^2+\mathrm{\cdots }+x_n^2}\sqrt{y_1^2+y_2^2+\mathrm{\cdots }+y_n^2}=\parallel 𝐱\parallel \parallel 𝐲\parallel ,$$

which implies

$${(x_1{y}_1+x_2{y}_2+\mathrm{\cdots }+x_n{y}_n)}^2\le \left(x_1^2+x_2^2+\mathrm{\cdots }+x_n^2\right)\left(y_1^2+y_2^2+\mathrm{\cdots }+y_n^2\right)$$

Notice that in this case inequality 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 http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Schwarz.htmlHermann Amandus Schwarz (1843–1921).

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

$\parallel 𝐱+𝐲\parallel \le \parallel 𝐱\parallel +\parallel 𝐲\parallel$ $\mathrm{\hspace{1em}\hspace{1em}}\text{triangle inequality}$

$|𝐱\cdot 𝐲|\le \parallel 𝐱\parallel \cdot \parallel 𝐲\parallel$ $\mathrm{\hspace{1em}\hspace{1em}}\text{CS inequality}$

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