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Concept
Polarization identity
Summary
In linear algebra, a branch of 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.
If a norm arises from an inner product then the polarization identity can be used to express this inner product entirely in terms of the norm. The polarization identity shows that a norm can arise from at most one inner product; however, there exist norms that do not arise from any inner product.
The norm associated with any inner product space satisfies the parallelogram law:
In fact, as observed by John von Neumann, the parallelogram law characterizes those norms that arise from inner products.
Given a normed space , the parallelogram law holds for if and only if there exists an inner product on such that for all in which case this inner product is uniquely determined by the norm via the polarization identity.
Any inner product on a vector space induces a norm by the equation
The polarization identities reverse this relationship, recovering the inner product from the norm.
Every inner product satisfies:
Solving for gives the formula If the inner product is real then and this formula becomes a polarization identity for real inner products.
If the vector space is over the real numbers then the polarization identities are:
These various forms are all equivalent by the parallelogram law:
This further implies that class is not a Hilbert space whenever , as the parallelogram law is not satisfied. For the sake of counterexample, consider and for any two disjoint subsets of general domain and compute the measure of both sets under parallelogram law.
For vector spaces over the complex numbers, the above formulas are not quite correct because they do not describe the imaginary part of the (complex) inner product.
However, an analogous expression does ensure that both real and imaginary parts are retained.
The complex part of the inner product depends on whether it is antilinear in the first or the second argument.
Official source
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