PlanetMath: Cauchy-Schwarz inequality

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Cauchy-Schwarz inequality

(Theorem)

Let be a vector space where an inner product $\langle,\rangle$ has been defined. Such spaces can be given also a norm by defining

$\displaystyle \Vert x\Vert = \sqrt{\langle x,x\rangle}.$

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

$\displaystyle \vert \langle v,w\rangle\vert\le \Vert v\Vert\Vert w\Vert$

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 $V=\mathbbmss{R}^n$ and the inner product is the dot product defined as $\langle v,w\rangle = v^t w$ and usually denoted as $v\cdot w$ and the resulting norm is the Euclidean norm.

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

$\displaystyle \vert \mathbf{x}\cdot \mathbf{y}\vert=\vert x_1y_1+x_2y_2+\cdots+... ...^2}\sqrt{y_1^2+y_2^2+\cdots+y_n^2}=\Vert \mathbf{x}\Vert\Vert \mathbf{y} \Vert,$

which implies

$\displaystyle (x_1y_1+x_2y_2+\cdots+x_ny_n)\sp2 \le\left(x_1^2+x_2^2+\cdots+x_n^2\right)\left( y_1^2+y_2^2+\cdots+y_n^2\right)$

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 othernames as Buniakovsky inequality or Kantorovich inequality. Another form that arises often is Cauchy-Schwartz inequality but this is a misspelling since the inequality is named after Hermann Amandus Schwarz (1843-1921).

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

$\displaystyle \Vert\mathbf{x}+\mathbf{y}\Vert\leq\Vert \mathbf{x}\Vert+\Vert \mathbf{y}\Vert$	triangle inequality
$\displaystyle \vert\mathbf{x}\cdot\mathbf{y}\vert\leq\Vert \mathbf{x}\Vert\cdot\Vert \mathbf{y}\Vert$	CS inequality

"Cauchy-Schwarz inequality" is owned by drini. [ full author list (2) | owner history (1) ]

(view preamble)

Other names:

Kantorovich inequality, Bunyakovsky inequality, Schwarz inequality, Cauchy inequality, CBS inequality

Also defines:

Cauchy-Schwartz inequality

Pronunciation (guide):

Cauchy-Schwarz inequality: /koh''''''''''''''''''''''''''''''''shee shvortz/

Attachments:: proof of Cauchy-Schwarz inequality (Proof) by mathwizard
proof of Cauchy-Schwarz inequality for real numbers (Proof) by stitch

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Cross-references: sums, triangle inequality, similar, fields, Hölder inequality, real number, term, even, inequality, implies, Euclidean norm, dot product, linearly dependent, equality, product, vectors, complex number, modulus, norm, inner product, vector space
There are 15 references to this entry.

This is version 15 of Cauchy-Schwarz inequality, born on 2002-02-01, modified 2006-09-22.
Object id is 1628, canonical name is CauchySchwarzInequality.
Accessed 97328 times total.

Classification:

AMS MSC:

15A63 (Linear and multilinear algebra; matrix theory :: Quadratic and bilinear forms, inner products)

Pending Errata and Addenda

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