polarization identity

The general claim can be argued using the polarization identity.

This is a special case of the polarization identity.

In the real case, the polarization identity is given by:

The polarization identity can be stated in the following way:

The polarization identities are not restricted to inner products.

In this case the standard polarization identities only give the real part of the inner product:

The covariation may be written in terms of the quadratic variation by the polarization identity:

Polarization identity, useful for rapid calculations involving Hessians.

If V is a complex vector space the inner product is given by the polarization identity:

The polarization identity can be generalized to various other contexts in abstract algebra, linear algebra, and functional analysis.

Another formula that appears to be as fast as Ryser's is closely related to the polarization identity for a symmetric tensor .

More generally, if f is any quadratic form, then the polarization of f agrees with the conclusion of the polarization identity.

Hadwiger's theorem implies the equivalence of Schläfli's stipulation and the geometrical definition of a eutactic star, by the polarization identity.

For any norm satisfying the parallelogram law (which necessarily is an inner product norm), the inner product generating the norm is unique as a consequence of the polarization identity.

One especially observes that for a complex Hilbert space, by way of the polarization identity, one easily verifies that Strong Operator convergence implies Weak Operator convergence.

In mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.

To see that it is sufficient—that the parallelogram law implies that the form defined by the polarization identity is indeed a complete inner product—one verifies algebraically that this form is additive, whence it follows mathematical induction that the form is linear over the integers and rationals.