# Polarization identity

﻿
Polarization identity

In mathematics, and more specifically in the theory of normed spaces and pre-Hilbert spaces in functional analysis, a vector space over the real numbers (the formula for the complex case is given in the article on Banach spaces) whose norm is defined in terms of its inner product satisfies as a necessary condition the polarization identity::$| x + y |^2 = |x|^2 + |y|^2 + 2langle x, y angle. qquad qquad \left(1\right)$

This identity is analogous to the formula for the square of a binomial:

:$\left(x + y\right)^2 = x^2 + y^2 + 2 x y. qquad qquad \left(2\right)$

If "y" in equation (1) is replaced by "−y" the result is:$| x - y |^2 = |x|^2 + |y|^2 - 2langle x, y angle. qquad qquad \left(3\right)$which corresponds to the cosine law and is analogous to equation (2) with "y" replaced by "−y":

:$\left(x - y\right)^2 = x^2 + y^2 - 2 x y. qquad qquad \left(4\right)$

Adding equations (1) and (3) yields

:$|x + y|^2 + |x - y|^2 = 2|x|^2 + 2|y|^2$

which corresponds to the parallelogram law and is analogous to the sum of equations (2) and (4):

:$\left(x + y\right)^2 + \left(x - y\right)^2 = 2 x^2 + 2 y^2.$

Subtracting (1) and (3) on the other hand gives

:$|x + y|^2 - |x - y|^2 = 4langle x,y angle. qquad$

Derivation

Let the norm of a vector be defined as the square root of the inner product of a vector with itself, like so

:$|x| = sqrt\left\{langle x, x angle\right\}. qquad qquad \left(5\right)$

Now find the inner product of "x" + "y" with itself:

:$langle x + y, x + y angle = langle x + y, x angle + langle x + y, y angle qquad qquad \left(6\right)$

is the result of distributing the first factor with respect to the sum of the second factor, which is possible due to linearity of the inner product. Distributing the second factors with respect to the sums of the first factors on the right side of equation (6) yields

:$langle x + y, x + y angle = langle x, x angle + langle y, x angle + langle x, y angle + langle y, y angle qquad qquad \left(7\right)$

and since the inner product is commutative, eq. (7) simplifies to

:$langle x + y, x + y angle = langle x, x angle + langle y, y angle + 2langle x, y angle. qquad qquad \left(8\right)$

Applying the definition of norm in equation (5) to equation (8), we obtain equation (1): the polarization identity.

Arbitrary rings

The polarization identity defined above applies to quadratic forms and the associated symmetric bilinear forms over any ring where 2 is invertible (such as $mathbf\left\{Q\right\}$ or any field of characteristic not equal to two,but not $mathbf\left\{Z\right\}$ or $mathbf\left\{F\right\}_2$), and shows that over such a ring, a quadratic form is equivalent to a bilinear form.

Where 2 is not invertible (such as over the integers), there is a difference, which is important for instance in L-theory.

Generalization

The identities may be extended more generally to homogeneous polynomials (that is, algebraic forms) of arbitrary degree, where it is known as the polarization formula, and is reviewed in greater detail in the article on the polarization of an algebraic form.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Polarization of an algebraic form — In mathematics, in particular in algebra, polarization is a technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables. Specifically, given a homogeneous polynomial, polarization produces a multilinear form …   Wikipedia

• Polarization — ( Brit. polarisation) is a property of waves that describes the orientation of their oscillations. For transverse waves, it describes the orientation of the oscillations in the plane perpendicular to the wave s direction of travel. Longitudinal… …   Wikipedia

• Identity crisis (psychology) — Erik Erikson, the psychologist who coined the term identity crisis, believes that the identity crisis is the most important conflict human beings encounter when they go through eight developmental stages in life.DescriptionThe identity is a… …   Wikipedia

• Ward-Takahashi identity — In quantum field theory, a Ward Takahashi identity is an identity between correlation functions that follows from the global or gauged symmetries of the theory, and which remains valid after renormalization. The Ward Takahashi identity of quantum …   Wikipedia

• Photon polarization — is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. Individual photons are completely polarized. Their polarization state can be linear or circular, or it can be elliptical, which is anywhere in …   Wikipedia

• Vacuum polarization — In quantum field theory, and specifically quantum electrodynamics, vacuum polarization describes a process in which a background electromagnetic field produces virtual electron positron pairs that change the distribution of charges and currents… …   Wikipedia

• Hilbert space — For the Hilbert space filling curve, see Hilbert curve. Hilbert spaces can be used to study the harmonics of vibrating strings. The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It… …   Wikipedia

• Banach space — In mathematics, Banach spaces (pronounced [ˈbanax]) is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every… …   Wikipedia

• List of mathematics articles (P) — NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… …   Wikipedia

• Clifford algebra — In mathematics, Clifford algebras are a type of associative algebra. They can be thought of as one of the possible generalizations of the complex numbers and quaternions.[1][2] The theory of Clifford algebras is intimately connected with the… …   Wikipedia