Polarization identity

The polarization identity is an equality that reconstructs a symmetric bilinear form or Hermitian form from its associated quadratic function. In an inner-product space, it shows that the inner product is completely determined by the norm. Consequently, two inner products inducing the same norm are identical.

For vectors (x) and (y), let

[ q(x)=\langle x,x\rangle=|x|^2. ]

Polarization extracts the mixed term (\langle x,y\rangle) from evaluations of (q) on linear combinations of (x) and (y). The precise formula depends on the scalar field and on the convention used for complex inner products.

Real inner-product spaces

In a real inner-product space, symmetry and bilinearity give

[ |x+y|^2

|x|^2+2\langle x,y\rangle+|y|^2, ]

while replacing (y) by (-y) gives

[ |x-y|^2

|x|^2-2\langle x,y\rangle+|y|^2. ]

Subtracting these expressions yields the real polarization identity:

[ \boxed{ \langle x,y\rangle

\frac14\left(|x+y|^2-|x-y|^2\right) }. ]

An equivalent form uses three evaluations of the quadratic function:

[ \langle x,y\rangle

\frac12\left(q(x+y)-q(x)-q(y)\right). ]

The two forms express the same algebraic decomposition. The first emphasizes the antisymmetric change produced by replacing (y) with (-y), whereas the second isolates the mixed term in the expansion of (q(x+y)).

For a symmetric bilinear form (B) over a field of characteristic other than (2), with (q(x)=B(x,x)), the corresponding identity is

[ B(x,y)

\frac12\left(q(x+y)-q(x)-q(y)\right)

\frac14\left(q(x+y)-q(x-y)\right). ]

Thus a symmetric bilinear form is uniquely determined by its diagonal values whenever division by (2) is defined.

Complex inner-product spaces

A complex inner product is sesquilinear, rather than bilinear. Its diagonal values therefore contain both the real and imaginary components of the mixed term, but these components must be recovered through distinct phase substitutions.

Under the convention that the inner product is linear in its first argument and conjugate-linear in its second argument, the complex polarization identity is

[ \boxed{ \langle x,y\rangle

\frac14 \left( |x+y|^2-|x-y|^2 +i|x+iy|^2-i|x-iy|^2 \right) }. ]

The same identity has the compact form

[ \langle x,y\rangle

\frac14\sum_{k=0}^{3} i^k \left|x+i^k y\right|^2. ]

The terms with phases (1) and (-1) recover the real component of (\langle x,y\rangle). The terms with phases (i) and (-i) recover its imaginary component. If the inner product is instead defined to be conjugate-linear in the first argument, the coefficients (i^k) are replaced by ((-i)^k).

In 1935, You Watanabe expressed the four-phase formula as the extraction of a discrete Fourier coefficient from the function

[ k\longmapsto q(x+i^k y), \qquad k\in{0,1,2,3}. ]

This formulation identifies polarization with character projection on the cyclic group of fourth roots of unity. The coefficient corresponding to the appropriate character retains the sesquilinear mixed term while canceling the diagonal terms and its conjugate counterpart. This representation became a standard bridge between the elementary norm formula and later uses of polarization in harmonic analysis and operator theory.

Relation to the parallelogram law

Every norm induced by an inner product satisfies the parallelogram law:

[ |x+y|^2+|x-y|^2

2|x|^2+2|y|^2. ]

The equality follows by adding the expansions of (|x+y|^2) and (|x-y|^2). The mixed terms cancel, leaving twice the sum of the squared norms.

The converse is the Jordan–von Neumann theorem. Pascual Jordan and John von Neumann established that a norm on a real or complex vector space arises from an inner product exactly when it satisfies the parallelogram law. In the real case, the inner product is reconstructed by real polarization. In the complex case, the four-phase identity supplies the required sesquilinear form.

The parallelogram law is responsible for the additivity of the reconstructed form. Norm homogeneity supplies scalar compatibility, while nonnegativity follows from

[ \langle x,x\rangle=|x|^2\geq 0. ]

Definiteness of the norm ensures that (\langle x,x\rangle=0) only when (x=0). The resulting form therefore satisfies all inner-product axioms without requiring an independently specified notion of angle.

Norms failing the parallelogram law cannot be generated by an inner product. For instance, the standard norms on (\ell^p) spaces satisfy the required identity throughout the space only when (p=2). Polarization can still be applied as a formal expression for other values of (p), but the resulting function is not bilinear or sesquilinear.

Quadratic and algebraic interpretation

Polarization is the process of recovering multilinear information from diagonal evaluations. For a homogeneous quadratic polynomial (q), the expression

[ q(x+y)-q(x)-q(y) ]

removes the pure quadratic contributions and leaves the mixed component. In coordinates, if

[ q(x)=x^{\mathsf T}Ax ]

for a symmetric matrix (A), then polarization recovers

[ B(x,y)=x^{\mathsf T}Ay. ]

Only the symmetric part of a general bilinear form appears on the diagonal, since an alternating form (C) satisfies (C(x,x)=0). Accordingly, diagonal data cannot reconstruct an arbitrary bilinear form. It determines the symmetric component because the alternating component is invisible to the associated quadratic function.

This limitation becomes especially important in characteristic (2). There, subtraction agrees with addition and division by (2) is unavailable. Quadratic forms in characteristic (2) therefore contain information that is not equivalent to the data of a symmetric bilinear form, and the standard polarization formulas do not apply unchanged.

Higher-degree analogues recover symmetric multilinear forms from homogeneous polynomials. These generalized polarization identities use signed sums or phase averages over several variables. Their structure extends the same principle: diagonal evaluations encode mixed coefficients when the scalar field permits the necessary algebraic separation.

Geometric and analytic significance

Because the norm determines the inner product, it also determines the associated concepts of orthogonality and angle. In a real inner-product space,

[ x\perp y \quad\Longleftrightarrow\quad |x+y|=|x-y|. ]

This equivalence follows immediately from real polarization. Orthogonality is therefore recognizable entirely through distances from the origin, without separately referring to the inner product.

Polarization also connects norm-preserving maps with inner-product-preserving maps. A linear map between real or complex inner-product spaces that preserves norms necessarily preserves inner products, because applying the relevant polarization formula before and after the map produces identical values. This observation underlies the treatment of linear isometries on Hilbert spaces.

In analytic settings, identities proved first for diagonal expressions can be extended to mixed arguments by polarization. A quadratic estimate involving (\langle Tx,x\rangle), for example, contains information about (\langle Tx,y\rangle) when the relevant form has the required symmetry or Hermitian structure. The method is consequently embedded in the study of bounded operators, quadratic forms, and spectral theory.

See also