Polarization identity explained

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:

\|x + y\| ^{2} + \|x - y\| ^{2} = 2 \|x\| ^{2} + 2 \|y\| ^{2.} In fact, as observed by John von Neumann, the parallelogram law characterizes those norms that arise from inner products. Given a normed space (H, \| ⋅ \|) , the parallelogram law holds for \| ⋅ \| if and only if there exists an inner product \langle ⋅ , ⋅ \rangle on H such that \|x\| ^{2} = \langle x, x\rangle for all x \in H, in which case this inner product is uniquely determined by the norm via the polarization identity.^{[1] [2]}

Polarization identities

Any inner product on a vector space induces a norm by the equation$$\backslash |x\backslash |\; =\; \backslash sqrt.$$The polarization identities reverse this relationship, recovering the inner product from the norm.Every inner product satisfies:$$\backslash |x\; +\; y\backslash |^2\; =\; \backslash |x\backslash |^2\; +\; \backslash |y\backslash |^2\; +\; 2\backslash operatorname\backslash langle\; x,\; y\; \backslash rangle\; \backslash qquad\; \backslash text\; x,\; y.$$

Solving for

\operatorname{Re}\langle x, y \rangle gives the formula \operatorname{Re}\langle x, y \rangle = (1)/(2) \left(\|x + y\| ^{2} - \|x\| ^{2} - \|y\| ^{2\right).} If the inner product is real then \operatorname{Re}\langle x, y \rangle = \langle x, y \rangle 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:

These various forms are all equivalent by the parallelogram law:$$2\backslash |x\backslash |^2\; +\; 2\backslash |y\backslash |^2\; =\; \backslash |x+y\backslash |^2\; +\; \backslash |x-y\backslash |^2.$$

This further implies that

L ^{p} class is not a Hilbert space whenever, as the parallelogram law is not satisfied. For the sake of counterexample, consider x =1 _{A} and y =1 _{B} for any two disjoint subsets A,B of general domain \Omega\subset R ^{n} 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

\langle x | y \rangle, which is commonly used in physics will be assumed to be antilinear in the argument while \langle x, y \rangle, which is commonly used in mathematics, will be assumed to be antilinear its the argument. They are related by the formula:$$\backslash langle\; x,\backslash ,\; y\; \backslash rangle\; =\; \backslash langle\; y\; \backslash ,|\backslash ,\; x\; \backslash rangle\; \backslash quad\; \backslash text\; x,\; y\; \backslash in\; H.$$

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

x, y \in H is always equal to:$$\backslash beginR(x,\; y)$$

&= \operatorname \langle x \mid y \rangle = \operatorname \langle x, y \rangle \\&= \frac \left(\|x+y\|^2 - \|x-y\|^2\right) \\&= \frac \left(\|x+y\|^2 - \|x\|^2 - \|y\|^2\right) \\[3pt]&= \frac \left(\|x\|^2 + \|y\|^2 - \|x-y\|^2\right). \\[3pt]\end

It is always a symmetric map, meaning that$$R(x,\; y)\; =\; R(y,\; x)\; \backslash quad\; \backslash text\; x,\; y\; \backslash in\; H,$$and it also satisfies: $$R(y,\; ix)\; =\; -\; R(x,\; iy)\; \backslash quad\; \backslash text\; x,\; y\; \backslash in\; H.$$Thus, which in plain English says that to move a factor of

i to the other argument, introduce a negative sign.

Let $$R(x,\; y)\; :=\; \backslash frac\; \backslash left(\backslash |x+y\backslash |^2\; -\; \backslash |x-y\backslash |^2\backslash right).$$ Then

2 \|x\| ^{2} + 2 \|y\| ^{2} = \|x + y\| ^{2} + \|x - y\| ^{2} implies $$R(x,\; y)\; =\; \backslash frac\; \backslash left(\backslash left(2\backslash |x\backslash |^2\; +\; 2\backslash |y\backslash |^2\; -\; \backslash |x-y\backslash |^2\backslash right)\; -\; \backslash |x-y\backslash |^2\backslash right)\; =\; \backslash frac\; \backslash left(\backslash |x\backslash |^2\; +\; \backslash |y\backslash |^2\; -\; \backslash |x-y\backslash |^2\backslash right)$$ and $$R(x,\; y)\; =\; \backslash frac\; \backslash left(\backslash |x+y\backslash |^2\; -\; \backslash left(2\backslash |x\backslash |^2\; +\; 2\backslash |y\backslash |^2\; -\; \backslash |x+y\backslash |^2\backslash right)\backslash right)\; =\; \backslash frac\; \backslash left(\backslash |x+y\backslash |^2\; -\; \backslash |x\backslash |^2\; -\; \backslash |y\backslash |^2\backslash right).$$

Moreover, $$4R(x,\; y)\; =\; \backslash |x+y\backslash |^2\; -\; \backslash |x-y\backslash |^2\; =\; \backslash |y+x\backslash |^2\; -\; \backslash |y-x\backslash |^2\; =\; 4R(y,\; x),$$ which proves that .

From

1 = i ( - i) it follows that y - ix = i( - iy - x) = - i(x + iy) and y + ix = i( - iy + x) = i(x - iy) so that $$-4R(y,\; ix)\; =\; \backslash |y-ix\backslash |^2\; -\; \backslash |y+ix\backslash |^2\; =\; \backslash |(-i)(x+iy)\backslash |^2\; -\; \backslash |i(x-iy)\backslash |^2\; =\; \backslash |x+iy\backslash |^2\; -\; \backslash |x-iy\backslash |^2\; =\; 4R(x,\; iy),$$ which proves that R(y, ix) = - R(x, iy). \blacksquare

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

\langle x | y \rangle, which is antilinear in the argument, are\begin{alignat}{ 4 } \langle x | y \rangle & = (1)/(4) \left(\|x + y\| ^{2} - \|x - y\| ^{2} - i\|x + iy\| ^{2} + i\|x - iy\| ^{2\right)} \\ & = R(x, y) - i R(x, iy) \\ & = R(x, y) + i R(ix, y) \\ \end{alignat} where x, y \in H. The second to last equality is similar to the formula expressing a linear functional \varphi in terms of its real part: \varphi(y) = \operatorname{Re} \varphi(y) - i (\operatorname{Re} \varphi)(i y).

Antilinear in second argument

The polarization identities for the inner product

\langle x, y \rangle, which is antilinear in the argument, follows from that of \langle x | y \rangle by the relationship: \langle x, y \rangle : = \langle y | x \rangle = \overline{\langle x | y \rangle} forall x, y \in H. So for any x, y \in H, \begin{alignat}{ 4 } \langle x, y \rangle & = (1)/(4) \left(\|x + y\| ^{2} - \|x - y\| ^{2} + i\|x + iy\| ^{2} - i\|x - iy\| ^{2\right)} \\ & = R(x, y) + i R(x, iy) \\ & = R(x, y) - i R(ix, y). \\ \end{alignat}

This expression can be phrased symmetrically as:^[3] $$\backslash langle\; x,\; y\; \backslash rangle\; =\; \backslash frac\; \backslash sum\_^3\; i^k\; \backslash left\backslash |x\; +\; i^k\; y\backslash right\backslash |^2.$$

Summary of both cases

Thus if

R(x, y) + i I(x, y) denotes the real and imaginary parts of some inner product's value at the point (x, y) \in H x H of its domain, then its imaginary part will be:where the scalar i is always located in the same argument that the inner product is antilinear in.

Using, the above formula for the imaginary part becomes:

Reconstructing the inner product

In a normed space

(H, \| ⋅ \|), if the parallelogram law$$\backslash |x+y\backslash |^2\; ~+~\; \backslash |x-y\backslash |^2\; ~=~\; 2\backslash |x\backslash |^2+2\backslash |y\backslash |^2$$holds, then there exists a unique inner product \langle ⋅ , ⋅ \rangle on H such that \|x\| ^{2} = \langle x, x\rangle for all x \in H.

Another necessary and sufficient condition for there to exist an inner product that induces a given norm

\| ⋅ \| is for the norm to satisfy Ptolemy's inequality, which is:^[4] $$\backslash |x\; -\; y\backslash |\; \backslash ,\; \backslash |z\backslash |\; ~+~\; \backslash |y\; -\; z\backslash |\; \backslash ,\; \backslash |x\backslash |\; ~\backslash geq~\; \backslash |x\; -\; z\backslash |\; \backslash ,\; \backslash |y\backslash |\; \backslash qquad\; \backslash text\; x,\; y,\; z.$$

Applications and consequences

If

H is a complex Hilbert space then \langle x \mid y \rangle is real if and only if its imaginary part is, which happens if and only if . Similarly, \langle x \mid y \rangle is (purely) imaginary if and only if . For example, from \|x + ix\| = | 1+ i| \|x\| = \sqrt{ 2 } \|x\| = | 1- i| \|x\| = \|x - ix\| it can be concluded that \langle x | x \rangle is real and that \langle x | ix \rangle is purely imaginary.

Isometries

If

A : H \to Z is a linear isometry between two Hilbert spaces (so \|A h\| = \|h\| for all h \in H ) then $$\backslash langle\; A\; h,\; A\; k\; \backslash rangle\_Z\; =\; \backslash langle\; h,\; k\; \backslash rangle\_H\; \backslash quad\; \backslash text\; h,\; k\; \backslash in\; H;$$that is, linear isometries preserve inner products.

If

A : H \to Z is instead an antilinear isometry then $$\backslash langle\; A\; h,\; A\; k\; \backslash rangle\_Z\; =\; \backslash overline\; =\; \backslash langle\; k,\; h\; \backslash rangle\_H\; \backslash quad\; \backslash text\; h,\; k\; \backslash in\; H.$$

Relation to the law of cosines

The second form of the polarization identity can be written as$$\backslash |\backslash textbf-\backslash textbf\backslash |^2\; =\; \backslash |\backslash textbf\backslash |^2\; +\; \backslash |\backslash textbf\backslash |^2\; -\; 2(\backslash textbf\; \backslash cdot\; \backslash textbf).$$

This is essentially a vector form of the law of cosines for the triangle formed by the vectors,, and . In particular,$$\backslash textbf\backslash cdot\backslash textbf\; =\; \backslash |\backslash textbf\backslash |\backslash ,\backslash |\backslash textbf\backslash |\; \backslash cos\backslash theta,$$where

\theta is the angle between the vectors bf{u} and .

The equation is numerically unstable if u and v are similar because of catastrophic cancellation and should be avoided for numeric computation.

Derivation

The basic relation between the norm and the dot product is given by the equation$$\backslash |\backslash textbf\backslash |^2\; =\; \backslash textbf\; \backslash cdot\; \backslash textbf.$$

Thenand similarly$$\backslash |\backslash textbf\; -\; \backslash textbf\backslash |^2\; =\; \backslash |\backslash textbf\backslash |^2\; +\; \backslash |\backslash textbf\backslash |^2\; -\; 2(\backslash textbf\; \backslash cdot\; \backslash textbf).$$

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

B is any symmetric bilinear form on a vector space, and Q is the quadratic form defined by$$Q(v)\; =\; B(v,\; v),$$then

The so-called symmetrization map generalizes the latter formula, replacing

Q by a homogeneous polynomial of degree k defined by Q(v) = B(v, \ldots, v), where B is a symmetric k -linear map.^[5]

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

B(u, v) 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

\varepsilon -quadratic forms and \varepsilon -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.

Notes and References

1. Book: Philippe Blanchard, Erwin Brüning. Proposition 14.1.2 (Fréchet–von Neumann–Jordan). https://books.google.com/books?id=1g2rikccHcgC&pg=PA192. 192. Mathematical methods in physics: distributions, Hilbert space operators, and variational methods. 2003. Birkhäuser. 0817642285.
2. Book: Gerald Teschl. Mathematical methods in quantum mechanics: with applications to Schrödinger operators. Theorem 0.19 (Jordan–von Neumann). 19. 978-0-8218-4660-5. 2009. American Mathematical Society Bookstore.
3. Web site: Butler. Jon. 20 June 2013. norm - Derivation of the polarization identities?. live. https://archive.today/20201014185358/https://math.stackexchange.com/questions/425173/derivation-of-the-polarization-identities. 14 October 2020. 2020-10-14. Mathematics Stack Exchange. See Harald Hanche-Olson's answer.
4. Apostol. Tom M.. 1967. Ptolemy's Inequality and the Chordal Metric. Mathematics Magazine. 40. 5. 233–235. en. 10.2307/2688275. 2688275.
5. . See Keith Conrad (KCd)'s answer.