- Polarization identity
In

mathematics , and more specifically in the theory ofnormed space s andpre-Hilbert space s infunctional analysis , avector 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 itsinner product satisfies as anecessary condition the**polarization identity**::$|\; x\; +\; y\; |^2\; =\; |x|^2\; +\; |y|^2\; +\; 2langle\; x,\; y\; angle.\; qquad\; qquad\; (1)$This identity is analogous to the formula for the square of a binomial:

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

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

cosine law and is analogous to equation (2) with "y" replaced by "−y"::$(x\; -\; y)^2\; =\; x^2\; +\; y^2\; -\; 2\; x\; y.\; qquad\; qquad\; (4)$

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

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

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

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

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

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

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

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

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

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 form s and the associated symmetricbilinear form s over any ring where 2 is invertible (such as $mathbf\{Q\}$ or any field of characteristic not equal to two,but not $mathbf\{Z\}$ or $mathbf\{F\}\_2$), 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 polynomial s (that is,algebraic form s) of arbitrary degree, where it is known as thepolarization formula , and is reviewed in greater detail in the article on thepolarization of an algebraic form .

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