Parallelogram law
The parallelogram law is an identity relating the lengths of the sides and diagonals of a parallelogram. In Euclidean geometry, it states that the sum of the squares of the four side lengths equals the sum of the squares of the two diagonal lengths. In the language of vector spaces, the same identity characterizes norms that arise from inner products.
For vectors (x) and (y) in an inner-product space, the law has the form
[ |x+y|^2+|x-y|^2
2|x|^2+2|y|^2. ]
Here (x+y) and (x-y) represent the diagonals of the parallelogram whose adjacent sides are represented by (x) and (y). The identity follows from the bilinearity or sesquilinearity of the inner product and does not depend on the dimension of the space.
Geometric formulation
Let a parallelogram have adjacent side lengths (a) and (b), with diagonal lengths (p) and (q). The parallelogram law states that
[ p^2+q^2=2a^2+2b^2. ]
This relation can be derived by dividing the parallelogram along either diagonal and applying the law of cosines. If (\theta) is the angle between adjacent sides, then the two diagonal lengths satisfy
[ p^2=a^2+b^2+2ab\cos\theta ]
and
[ q^2=a^2+b^2-2ab\cos\theta. ]
Adding these expressions eliminates the angular term and produces the parallelogram law. The cancellation shows that the combined squared length of the diagonals is independent of the angle between the sides when the two side lengths remain fixed.
A rectangle gives the special case in which both diagonals have the same length. The resulting equation reduces to two simultaneous instances of the Pythagorean theorem. A rhombus gives a different specialization, since equal side lengths imply that the sum of the squared diagonal lengths is four times the square of a side length.
Inner-product derivation
In a real inner-product space, the squared norm of a vector is defined by
[ |x|^2=\langle x,x\rangle. ]
Expansion of the two diagonal terms gives
[ |x+y|^2
\langle x,x\rangle +2\langle x,y\rangle +\langle y,y\rangle ]
and
[ |x-y|^2
\langle x,x\rangle -2\langle x,y\rangle +\langle y,y\rangle. ]
Their sum cancels the mixed terms, leaving twice the squared norm of each vector. In a complex inner-product space, conjugate symmetry changes the separate expansions by replacing the mixed contribution with twice the real part of (\langle x,y\rangle), but the same cancellation occurs.
The identity therefore holds in every Hilbert space, including infinite-dimensional spaces. Its validity depends on the inner-product structure rather than on coordinates, a selected basis, or a finite-dimensional geometric realization.
Characterization of inner-product norms
The converse of the vector identity is the Jordan–von Neumann theorem. It states that a norm on a real or complex vector space is induced by an inner product if and only if the norm satisfies the parallelogram law for every pair of vectors.
In the real case, the corresponding inner product is recovered through the polarization identity:
[ \langle x,y\rangle
\frac{1}{4} \left( |x+y|^2-|x-y|^2 \right). ]
For complex vector spaces, the norm determines the inner product through the formula
[ \langle x,y\rangle
\frac{1}{4} \left( |x+y|^2-|x-y|^2 +i|x+iy|^2-i|x-iy|^2 \right), ]
under the convention that the inner product is linear in its first argument. The conjugated formula applies under the opposite convention.
The parallelogram law supplies the algebraic compatibility needed for these expressions to be additive. Consequently, a general normed vector space need not admit an inner product that generates its norm. For example, the usual norm on an (L^p) space satisfies the identity in general only when (p=2).
Historical development
The geometric content of the law appears in classical Greek treatments of metric relations in parallelograms. Euclid expressed the relevant relation through propositions concerning parallelograms and their diagonals, while Apollonius of Perga developed a closely related theorem for the median of a triangle. The latter result, now called Apollonius's theorem, is equivalent to the parallelogram law after a triangle is completed to a parallelogram.
The modern functional-analytic characterization emerged from the study of abstract normed spaces during the early twentieth century. In 1935, Pascual Jordan, John von Neumann, and You Watanabe established the equivalence between the parallelogram identity and the existence of a norm-generating inner product. Their treatment connected the elementary geometric equation with the structural distinction between general normed spaces and inner-product spaces.
Subsequent work placed the characterization within the theory of Banach spaces. In that setting, the law separates Hilbert-space geometry from norms whose unit balls do not arise from positive-definite quadratic forms.
Quadratic-form interpretation
The squared norm in an inner-product space is a quadratic form,
[ Q(x)=|x|^2. ]
The parallelogram law then becomes
[ Q(x+y)+Q(x-y)=2Q(x)+2Q(y). ]
This equation is the quadratic analogue of additivity. When it is accompanied by the homogeneity properties of a norm, it permits the associated symmetric bilinear form to be reconstructed by polarization. In finite-dimensional real spaces, the resulting form can be represented by a positive-definite matrix (A), so that
[ |x|^2=x^{\mathsf T}Ax. ]
The parallelogram identity is preserved under linear changes of coordinates because it concerns the underlying quadratic form rather than a particular matrix representation.
The shape of the unit ball reflects the same distinction. A finite-dimensional inner-product norm has an ellipsoid as its unit sphere after an appropriate choice of coordinates, whereas a general norm may produce a unit sphere without quadratic geometry. The failure of the parallelogram law records this difference algebraically.
Relation to triangle medians
The law can also be stated as a relation involving a median of a triangle. If a triangle has side lengths (a), (b), and (c), and (m) is the length of the median to the side of length (c), then
[ a^2+b^2=2m^2+\frac{c^2}{2}. ]
Completing the triangle to a parallelogram identifies the median with half of one diagonal. This correspondence makes Apollonius's theorem and the parallelogram law equivalent formulations of the same Euclidean metric identity.