Polarization identity

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: formula_1 
In fact, as observed by John von Neumann, the parallelogram law characterizes those norms that arise from inner products. 
Given a normed space formula_2, the parallelogram law holds for formula_3 if and only if there exists an inner product formula_4 on formula_5 such that formula_6 for all formula_7 in which case this inner product is uniquely determined by the norm via the polarization identity.
Polarization identities.
Any inner product on a vector space induces a norm by the equation
formula_8
The polarization identities reverse this relationship, recovering the inner product from the norm.
Every inner product satisfies:
formula_9
Solving for formula_10 gives the formula formula_11 If the inner product is real then formula_12 and this formula becomes a polarization identity for real inner products.
Real vector spaces.
If the vector space is over the real numbers then the polarization identities are:
formula_13
These various forms are all equivalent by the parallelogram law:
formula_14
This further implies that formula_15 class is not a Hilbert space whenever formula_16, as the parallelogram law is not satisfied. For the sake of counterexample, consider formula_17 and formula_18 for any two disjoint subsets formula_19 of general domain formula_20 and compute the measure of both sets under parallelogram law.
Complex vector spaces.
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. 
The notation formula_21 which is commonly used in physics will be assumed to be antilinear in the argument while formula_22 which is commonly used in mathematics, will be assumed to be antilinear its the argument. 
They are related by the formula:
formula_23
The real part of any inner product (no matter which argument is antilinear and no matter if it is real or complex) is a symmetric bilinear map that for any formula_24 is always equal to:
formula_25
It is always a symmetric map, meaning that
formula_26
and it also satisfies: 
formula_27
Thus formula_28 which in plain English says that to move a factor of formula_29 to the other argument, introduce a negative sign. 
Let 
formula_30 
Then formula_31 implies 
formula_32 
and 
formula_33 
Moreover, 
formula_34 
which proves that formula_35 
From formula_36 it follows that formula_37 and formula_38 so that 
formula_39 
which proves that formula_40 
formula_41
Unlike its real part, the imaginary part of a complex inner product depends on which argument is antilinear. 
Antilinear in first argument
The polarization identities for the inner product formula_42 which is antilinear in the argument, are
where formula_44 
The second to last equality is similar to the formula expressing a linear functional formula_45 in terms of its real part: formula_46 
Antilinear in second argument
The polarization identities for the inner product formula_47 which is antilinear in the argument, follows from that of formula_48 by the relationship: 
formula_49 
So for any formula_50
This expression can be phrased symmetrically as:
formula_52
Summary of both cases
Thus if formula_53 denotes the real and imaginary parts of some inner product's value at the point formula_54 of its domain, then its imaginary part will be:
formula_55
where the scalar formula_56 is always located in the same argument that the inner product is antilinear in.
Using formula_28 the above formula for the imaginary part becomes:
formula_58
Reconstructing the inner product.
In a normed space formula_59 if the parallelogram law
formula_60
holds, then there exists a unique inner product formula_61 on formula_5 such that formula_6 for all formula_64
Another necessary and sufficient condition for there to exist an inner product that induces a given norm formula_3 is for the norm to satisfy Ptolemy's inequality, which is:
formula_66
Applications and consequences.
If formula_5 is a complex Hilbert space then formula_68 is real if and only if its imaginary part is formula_69 which happens if and only if formula_70 
Similarly, formula_68 is (purely) imaginary if and only if formula_72 
For example, from formula_73 it can be concluded that formula_74 is real and that formula_75 is purely imaginary. 
Isometries.
If formula_76 is a linear isometry between two Hilbert spaces (so formula_77 for all formula_78) then 
formula_79
that is, linear isometries preserve inner products. 
If formula_76 is instead an antilinear isometry then 
formula_81 
Relation to the law of cosines.
The second form of the polarization identity can be written as
formula_82
This is essentially a vector form of the law of cosines for the triangle formed by the vectors formula_83 and formula_84 
In particular,
formula_85
where formula_86 is the angle between the vectors formula_87 and formula_88
Derivation.
The basic relation between the norm and the dot product is given by the equation
formula_89
Then
formula_90
and similarly
formula_91
Forms (1) and (2) of the polarization identity now follow by solving these equations for formula_92 while form (3) follows from subtracting these two equations. 
Generalizations.
Symmetric bilinear forms.
The polarization identities are not restricted to inner products. 
If formula_93 is any symmetric bilinear form on a vector space, and formula_94 is the quadratic form defined by
formula_95
then
formula_96
The so-called symmetrization map generalizes the latter formula, replacing formula_94 by a homogeneous polynomial of degree formula_98 defined by formula_99 where formula_93 is a symmetric formula_98-linear map.
The formulas above even apply in the case where the field of scalars has characteristic two, though the left-hand sides are all zero in this case. 
Consequently, in characteristic two there is no formula for a symmetric bilinear form in terms of a quadratic form, and they are in fact distinct notions, a fact which has important consequences in L-theory; for brevity, in this context "symmetric bilinear forms" are often referred to as "symmetric forms".
These formulas also apply to bilinear forms on modules over a commutative ring, though again one can only solve for formula_102 if 2 is invertible in the ring, and otherwise these are distinct notions. For example, over the integers, one distinguishes integral quadratic forms from integral forms, which are a narrower notion.
More generally, in the presence of a ring involution or where 2 is not invertible, one distinguishes formula_103-quadratic forms and formula_103-symmetric forms; a symmetric form defines a quadratic form, and the polarization identity (without a factor of 2) from a quadratic form to a symmetric form is called the "symmetrization map", and is not in general an isomorphism. This has historically been a subtle distinction: over the integers it was not until the 1950s that relation between "twos out" (integral form) and "twos in" (integral form) was understood – see discussion at integral quadratic form; and in the algebraization of surgery theory, Mishchenko originally used "L"-groups, rather than the correct "L"-groups (as in Wall and Ranicki) – see discussion at L-theory.
Homogeneous polynomials of higher degree.
Finally, in any of these contexts these identities may be extended 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.