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
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:
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
holds, then there is an inner product on such that for all
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
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 positive-definite,
real-valued, -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.
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
where is the angle between the vectors and
The basic relation between the norm and the dot product is given by the equation
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.)
symmetrization map generalizes the latter formula, replacing by a homogeneous polynomial of degree defined by where is a symmetric k-linear map.
The formulas above even apply in the case where the
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 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 symmetricL-groups, rather than the correct quadraticL-groups (as in Wall and Ranicki) – see discussion at