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 (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 forms and the associated symmetric bilinear forms 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 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.


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