Parallelogram law

The sides of parallelogram ABCD are shown in blue and the diagonals in red. The sum of the areas of the blue squares equal that of the red ones.

In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. We use these notations for the sides: AB, BC, CD, DA. But since in Euclidean geometry a parallelogram necessarily has opposite sides equal, that is, AB = CD and BC = DA, the law can be stated as $2AB^{2}+2BC^{2}=AC^{2}+BD^{2}\,$

If the parallelogram is a rectangle, the two diagonals are of equal lengths AC = BD, so $2AB^{2}+2BC^{2}=2AC^{2}$ and the statement reduces to the Pythagorean theorem. For the general quadrilateral (with four sides not necessarily equal) Euler's quadrilateral theorem states $AB^{2}+BC^{2}+CD^{2}+DA^{2}=AC^{2}+BD^{2}+4x^{2},$ where $x$ is the length of the line segment joining the midpoints of the diagonals. It can be seen from the diagram that $x=0$ for a parallelogram, and so the general formula simplifies to the parallelogram law.

Proof

In the parallelogram on the right, let AD = BC = a, AB = DC = b, $\angle BAD=\alpha .$ By using the law of cosines in triangle $\triangle BAD,$ we get: $a^{2}+b^{2}-2ab\cos(\alpha )=BD^{2}.$

In a parallelogram, adjacent angles are supplementary, therefore $\angle ADC=180^{\circ }-\alpha .$ Using the law of cosines in triangle $\triangle ADC,$ produces: $a^{2}+b^{2}-2ab\cos(180^{\circ }-\alpha )=AC^{2}.$

By applying the trigonometric identity $\cos(180^{\circ }-x)=-\cos x$ to the former result proves: $a^{2}+b^{2}+2ab\cos(\alpha )=AC^{2}.$

Now the sum of squares $BD^{2}+AC^{2}$ can be expressed as: $BD^{2}+AC^{2}=a^{2}+b^{2}-2ab\cos(\alpha )+a^{2}+b^{2}+2ab\cos(\alpha ).$

Simplifying this expression, it becomes: $BD^{2}+AC^{2}=2a^{2}+2b^{2}.$

The parallelogram law in inner product spaces

Vectors involved in the parallelogram law.

In a normed space, the statement of the parallelogram law is an equation relating norms: $2\|x\|^{2}+2\|y\|^{2}=\|x+y\|^{2}+\|x-y\|^{2}\quad {\text{ for all }}x,y.$

The parallelogram law is equivalent to the seemingly weaker statement: $2\|x\|^{2}+2\|y\|^{2}\leq \|x+y\|^{2}+\|x-y\|^{2}\quad {\text{ for all }}x,y$ because the reverse inequality can be obtained from it by substituting ${\textstyle {\frac {1}{2}}\left(x+y\right)}$ for $x,$ and ${\textstyle {\frac {1}{2}}\left(x-y\right)}$ for $y,$ and then simplifying. With the same proof, the parallelogram law is also equivalent to: $\|x+y\|^{2}+\|x-y\|^{2}\leq 2\|x\|^{2}+2\|y\|^{2}\quad {\text{ for all }}x,y.$

In an inner product space, the norm is determined using the inner product: $\|x\|^{2}=\langle x,x\rangle .$

As a consequence of this definition, in an inner product space the parallelogram law is an algebraic identity, readily established using the properties of the inner product: $\|x+y\|^{2}=\langle x+y,x+y\rangle =\langle x,x\rangle +\langle x,y\rangle +\langle y,x\rangle +\langle y,y\rangle ,$ $\|x-y\|^{2}=\langle x-y,x-y\rangle =\langle x,x\rangle -\langle x,y\rangle -\langle y,x\rangle +\langle y,y\rangle .$

Adding these two expressions: $\|x+y\|^{2}+\|x-y\|^{2}=2\langle x,x\rangle +2\langle y,y\rangle =2\|x\|^{2}+2\|y\|^{2},$ as required.

If $x$ is orthogonal to $y,$ meaning $\langle x,\ y\rangle =0,$ and the above equation for the norm of a sum becomes: $\|x+y\|^{2}=\langle x,x\rangle +\langle x,y\rangle +\langle y,x\rangle +\langle y,y\rangle =\|x\|^{2}+\|y\|^{2},$ which is Pythagoras' theorem.

Normed vector spaces satisfying the parallelogram law

Most real and complex normed vector spaces do not have inner products, but all normed vector spaces have norms (by definition). For example, a commonly used norm for a vector $x=(x_{1},x_{2},\ldots ,x_{n})$ in the real coordinate space $\mathbb {R} ^{n}$ is the $p$ -norm: $\|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dotsb +|x_{n}|^{p}\right)^{1/p}.$

Given a norm, one can evaluate both sides of the parallelogram law above. A remarkable fact is that if the parallelogram law holds, then the norm must arise in the usual way from some inner product. In particular, it holds for the $p$ -norm if and only if $p=2,$ the so-called Euclidean norm or standard norm.^[1]^[2]

For any norm satisfying the parallelogram law (which necessarily is an inner product norm), the inner product generating the norm is unique as a consequence of the polarization identity. In the real case, the polarization identity is given by: $\langle x,y\rangle ={\frac {\|x+y\|^{2}-\|x-y\|^{2}}{4}},$ or equivalently by ${\frac {\|x+y\|^{2}-\|x\|^{2}-\|y\|^{2}}{2}}\qquad {\text{ or }}\qquad {\frac {\|x\|^{2}+\|y\|^{2}-\|x-y\|^{2}}{2}}.$

In the complex case it is given by: $\langle x,y\rangle ={\frac {\|x+y\|^{2}-\|x-y\|^{2}}{4}}+i{\frac {\|ix-y\|^{2}-\|ix+y\|^{2}}{4}}.$

For example, using the $p$ -norm with $p=2$ and real vectors $x$ and $y,$ the evaluation of the inner product proceeds as follows: ${\begin{aligned}\langle x,y\rangle &={\frac {\|x+y\|^{2}-\|x-y\|^{2}}{4}}\\[4mu]&={\tfrac {1}{4}}\left(\sum _{i}|x_{i}+y_{i}|^{2}-\sum _{i}|x_{i}-y_{i}|^{2}\right)\\[2mu]&={\tfrac {1}{4}}\left(4\sum _{i}x_{i}y_{i}\right)\\&=x\cdot y,\\\end{aligned}}$ which is the standard dot product of two vectors.

Another necessary and sufficient condition for there to exist an inner product that induces the given norm $\|\cdot \|$ is for the norm to satisfy Ptolemy's inequality:^[3] $\|x-y\|\,\|z\|~+~\|y-z\|\,\|x\|~\geq ~\|x-z\|\,\|y\|\qquad {\text{ for all vectors }}x,y,z.$

References

^ Cantrell, Cyrus D. (2000). Modern mathematical methods for physicists and engineers. Cambridge University Press. p. 535. ISBN 0-521-59827-3. if p ≠ 2, there is no inner product such that ${\textstyle {\sqrt {\langle x,\ x\rangle }}=\|x\|_{p}}$ because the p-norm violates the parallelogram law.
^ Saxe, Karen (2002). Beginning functional analysis. Springer. p. 10. ISBN 0-387-95224-1.
^ Apostol, Tom M. (1967). "Ptolemy's Inequality and the Chordal Metric". Mathematics Magazine. 40 (5): 233–235. doi:10.2307/2688275. JSTOR 2688275.

External links

[Pelinovsky-1] Cantrell, Cyrus D. (2000). Modern mathematical methods for physicists and engineers. Cambridge University Press. p. 535. ISBN 0-521-59827-3. if p ≠ 2, there is no inner product such that ${\textstyle {\sqrt {\langle x,\ x\rangle }}=\|x\|_{p}}$ because the p-norm violates the parallelogram law.

[Saxe-2] Saxe, Karen (2002). Beginning functional analysis. Springer. p. 10. ISBN 0-387-95224-1.

[3] Apostol, Tom M. (1967). "Ptolemy's Inequality and the Chordal Metric". Mathematics Magazine. 40 (5): 233–235. doi:10.2307/2688275. JSTOR 2688275.

[1]

[2]

[3]

v t e Hilbert spaces
Basic concepts	Adjoint Inner product and L-semi-inner product Hilbert space and Prehilbert space Orthogonal complement Orthonormal basis
Main results	Bessel's inequality Cauchy–Schwarz inequality Riesz representation
Other results	Hilbert projection theorem Parseval's identity Polarization identity (Parallelogram law)
Maps	Compact operator on Hilbert space Densely defined Hermitian form Hilbert–Schmidt Normal Self-adjoint Sesquilinear form Trace class Unitary
Examples	Cⁿ(K) with K compact & n<∞ Segal–Bargmann F

v t e Banach space topics
Types of Banach spaces	Asplund Banach list Banach lattice Grothendieck Hilbert Inner product space Polarization identity (Polynomially) Reflexive Riesz L-semi-inner product (B Strictly Uniformly) convex Uniformly smooth (Injective Projective) Tensor product (of Hilbert spaces)
Banach spaces are:	Barrelled Complete F-space Fréchet tame Locally convex Seminorms/Minkowski functionals Mackey Metrizable Normed norm Quasinormed Stereotype
Function space Topologies	Banach–Mazur compactum Dual Dual space Dual norm Operator Ultraweak Weak polar operator Strong polar operator Ultrastrong Uniform convergence
Linear operators	Adjoint Bilinear form operator sesquilinear (Un)Bounded Closed Compact on Hilbert spaces (Dis)Continuous Densely defined Fredholm kernel operator Hilbert–Schmidt Functionals positive Pseudo-monotone Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary
Operator theory	Banach algebras C-algebras Operator space Spectrum C-algebra radius Spectral theory of ODEs Spectral theorem Polar decomposition Singular value decomposition
Theorems	Anderson–Kadec Banach–Alaoglu Banach–Mazur Banach–Saks Banach–Schauder (open mapping) Banach–Steinhaus (Uniform boundedness) Bessel's inequality Cauchy–Schwarz inequality Closed graph Closed range Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Gelfand–Naimark Goldstine Hahn–Banach hyperplane separation Kakutani fixed-point Krein–Milman Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Parseval's identity Riesz's lemma Riesz representation Robinson-Ursescu Schauder fixed-point
Analysis	Abstract Wiener space Banach manifold bundle Bochner space Convex series Differentiation in Fréchet spaces Derivatives Fréchet Gateaux functional holomorphic quasi Integrals Bochner Dunford Gelfand–Pettis regulated Paley–Wiener weak Functional calculus Borel continuous holomorphic Measures Lebesgue Projection-valued Vector Weakly / Strongly measurable function
Types of sets	Absolutely convex Absorbing Affine Balanced/Circled Bounded Convex Convex cone (subset) Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx)) Linear cone (subset) Radial Radially convex/Star-shaped Symmetric Zonotope
Subsets / set operations	Affine hull (Relative) Algebraic interior (core) Bounding points Convex hull Extreme point Interior Linear span Minkowski addition Polar (Quasi) Relative interior
Examples	Absolute continuity AC $ba(\Sigma )$ c space Banach coordinate BK Besov $B_{p,q}^{s}(\mathbb {R} )$ Birnbaum–Orlicz Bounded variation BV Bs space Continuous C(K) with K compact Hausdorff Hardy H^p Hilbert H Morrey–Campanato $L^{\lambda ,p}(\Omega )$ ℓ^p $\ell ^{\infty }$ L^p $L^{\infty }$ weighted Schwartz $S\left(\mathbb {R} ^{n}\right)$ Segal–Bargmann F Sequence space Sobolev W^k,p Sobolev inequality Triebel–Lizorkin Wiener amalgam $W(X,L^{p})$
Applications	Differential operator Finite element method Mathematical formulation of quantum mechanics Ordinary Differential Equations (ODEs) Validated numerics