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. Equivalently, the polarization identity describes when a norm can be assumed to arise from an inner product. In that terminology:^{ [1]}^{ [2]}
The norm associated with any inner product space satisfies the parallelogram law. In fact, the parallelogram law characterizes those normed spaces that arise from inner product spaces. Explicitly, this means that a normed space the parallelogram law holds if and only if there exists an inner product on such that for all
Formulas
Any inner product on a vector space induces a norm by the equation
Real vector spaces
If the vector space is over the real numbers, then expanding out the squares of binomials reveals
These various forms are all equivalent by the 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 which is commonly used in physics will be assumed to be antilinear in the first argument while which is commonly used in mathematics, will be assumed to be antilinear its the second argument. They are related by the formula:
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 is always equal to:
The equalities and hold for all vectors
 Antilinear in first argument
For the inner product which is antilinear in the first argument, for any
The second to last equality is similar to the formula expressing a linear functional in terms of its real part:
 Antilinear in second argument
The formula for the inner product which is antilinear in the second argument, follows from that of by the relationship: So for any
This expression can be phrased symmetrically as:^{ [3]}
Reconstructing the inner product
In a normed space if the parallelogram law
Proof


We will only give the real case here; the proof for complex vector spaces is analogous. By the above formulas, if the norm is described by an inner product (as we hope), then it must satisfy We need prove that this formula defines an inner product which induces the norm That is, we must show: (This axiomatization omits positivity, which is implied by (1) and the fact that is a norm.) For properties (1) and (2), substitute: and For property (3), it is convenient to work in reverse. It remains to show that
or equivalently,
Now apply the parallelogram identity: Thus it remains to verify: But the latter claim can be verified by subtracting the following two further applications of the parallelogram identity: Thus (3) holds. It can be verified by induction that (3) implies (4), as long as But "(4) when " implies "(4) when ". And any positivedefinite, realvalued, bilinear form satisfies the Cauchy–Schwarz inequality, so that is continuous. Thus must be linear as well. 
Applications and consequences
If is a complex Hilbert space then is real if and only if its complex part is which happens if and only if Similarly, is (purely) imaginary if and only if For example, from it can be concluded that is real and that is purely imaginary.
Isometries
If is a linear isometry between two Hilbert spaces (so for all ) then
If is instead an antilinear isometry then
Relation to the law of cosines
The second form of the polarization identity can be written as
This is essentially a vector form of the law of cosines for the triangle formed by the vectors and In particular,
where is the angle between the vectors and
Derivation
The basic relation between the norm and the dot product is given by the equation
Then
Forms (1) and (2) of the polarization identity now follow by solving these equations for while form (3) follows from subtracting these two equations. (Adding these two equations together gives the parallelogram law.)
Generalizations
Symmetric bilinear forms
The polarization identities are not restricted to inner products. If is any symmetric bilinear form on a vector space, and is the quadratic form defined by
The socalled symmetrization map generalizes the latter formula, replacing by a homogeneous polynomial of degree defined by where is a symmetric klinear map.^{ [4]}
The formulas above even apply in the case where the field of scalars has characteristic two, though the lefthand 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 Ltheory; 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 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 symmetric forms, which are a narrower notion.
More generally, in the presence of a ring involution or where 2 is not invertible, one distinguishes quadratic forms and 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 quadratic form) and "twos in" (integral symmetric form) was understood – see discussion at integral quadratic form; and in the algebraization of surgery theory, Mishchenko originally used symmetric Lgroups, rather than the correct quadratic Lgroups (as in Wall and Ranicki) – see discussion at Ltheory.
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.
See also
 Inner product space – Generalization of the dot product; used to define Hilbert spaces
 Law of cosines
 Minkowski distance
 Parallelogram law – The sum of the squares of the 4 sides of a parallelogram equals that of the 2 diagonals
Notes and references
 ^ Philippe Blanchard, Erwin Brüning (2003). "Proposition 14.1.2 (Fréchet–von Neumann–Jordan)". Mathematical methods in physics: distributions, Hilbert space operators, and variational methods. Birkhäuser. p. 192. ISBN 0817642285.
 ^ Gerald Teschl (2009). "Theorem 0.19 (Jordan–von Neumann)". Mathematical methods in quantum mechanics: with applications to Schrödinger operators. American Mathematical Society Bookstore. p. 19. ISBN 9780821846605.
 ^ Butler, Jon (20 June 2013). "norm  Derivation of the polarization identities?". Mathematics Stack Exchange. Archived from the original on 14 October 2020. Retrieved 20201014. See Harald HancheOlson's answer.
 ^ Butler 2013. See Keith Conrad (KCd)'s answer.
Bibliography
 Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGrawHill Science/Engineering/Math. ISBN 9780070542365. OCLC 21163277.