Parallelogram identity

Parallelogram

The parallelogram identity is one of the equalities in vector algebra and vector analysis .

Content

In Euclidean geometry

The sum of the squared lengths of the sides of the parallelogram is the sum of the squared lengths of its diagonals .

\ (AB)^{2}+(BC)^{2}+(CD)^{2}+(DA)^{2}=(AC)^{2}+(BD)^{2}.

{\ displaystyle \ (AB) ^ {2} + (BC) ^ {2} + (CD) ^ {2} + (DA) ^ {2} = (AC) ^ {2} + (BD) ^ {2 }.}

In spaces with scalar product

Parallelogram Identity Illustration

In vector spaces with a scalar product, this identity looks like this ^[1] :

\ 2\|x\|^{2}+2\|y\|^{2}=\|x+y\|^{2}+\|x-y\|^{2}

{\ displaystyle \ 2 \ | x \ | ^ {2} +2 \ | y \ | ^ {2} = \ | x + y \ | ^ {2} + \ | xy \ | ^ {2}}

Where

\ \|x\|^{2}=\langle x,x\rangle .

{\ displaystyle \ \ | x \ | ^ {2} = \ langle x, x \ rangle.}

In normed spaces (polarization identity)

In the normed space ( V , $\|\cdot \|$ ${\ displaystyle \ | \ cdot \ |}$ ), for which the parallelogram identity holds, we can introduce the scalar product $\langle x,\ y\rangle$ ${\ displaystyle \ langle x, \ y \ rangle}$ generating this norm, that is, such that $\|x\|^{2}=\langle x,\ x\rangle$ ${\ displaystyle \ | x \ | ^ {2} = \ langle x, \ x \ rangle}$ all vectors $x$ ${\ displaystyle x}$ of space $V$ ${\ displaystyle V}$ . This theorem is attributed to Frechet , von Neumann and Jordan ^[2] ^[3] . This can be done in the following way:

for real space
$\langle x,y\rangle ={\|x+y\|^{2}-\|x-y\|^{2} \over 4},$ ${\ displaystyle \ langle x, y \ rangle = {\ | x + y \ | ^ {2} - \ | xy \ | ^ {2} \ over 4},}$ or ${\|x+y\|^{2}-\|x\|^{2}-\|y\|^{2} \over 2},$ ${\ displaystyle {\ | x + y \ | ^ {2} - \ | x \ | ^ {2} - \ | y \ | ^ {2} \ over 2},}$ or ${\|x\|^{2}+\|y\|^{2}-\|x-y\|^{2} \over 2}.$ ${\ displaystyle {\ | x \ | ^ {2} + \ | y \ | ^ {2} - \ | xy \ | ^ {2} \ over 2}.}$
for complex space
$\langle x,y\rangle ={\|x+y\|^{2}-\|x-y\|^{2} \over 4}+i{\|ix-y\|^{2}-\|ix+y\|^{2} \over 4}.$ ${\ displaystyle \ langle x, y \ rangle = {\ | x + y \ | ^ {2} - \ | xy \ | ^ {2} \ over 4} + i {\ | ix-y \ | ^ {2 } - \ | ix + y \ | ^ {2} \ over 4}.}$

The above formulas expressing the scalar product of two vectors in terms of the norm are called the polarization identity .

Obviously, the norm expressed in terms of any scalar product as follows $\ \|x\|^{2}=\langle x,x\rangle$ ${\ displaystyle \ \ | x \ | ^ {2} = \ langle x, x \ rangle}$ will satisfy this identity.

Polarization identity is often used to turn Banach spaces into Hilbert spaces .

Summary

If B is a symmetric bilinear form in a vector space, and the quadratic form Q is expressed as

\ Q(v)=B(v,v)

{\ displaystyle \ Q (v) = B (v, v)}

,

then

{\begin{array}{l}4B(u,v)=Q(u+v)-Q(u-v),\\2B(u,v)=Q(u+v)-Q(u)-Q(v),\\2B(u,v)=Q(u)+Q(v)-Q(u-v).\end{array}}

{\ displaystyle {\ begin {array} {l} 4B (u, v) = Q (u + v) -Q (uv), \\ 2B (u, v) = Q (u + v) -Q (u ) -Q (v), \\ 2B (u, v) = Q (u) + Q (v) -Q (uv). \ End {array}}}

Notes

↑ Shilov, 1961 , p. 185.
↑ Philippe Blanchard, Erwin Brüning. Proposition 14.1.2 (Fréchet – von Neumann – Jordan) // Mathematical methods in physics: distributions, Hilbert space operators, and variational methods . - Birkhäuser, 2003 .-- P. 192. - ISBN 0817642285 .
↑ Gerald Teschl. Theorem 0.19 (Jordan – von Neumann) // Mathematical methods in quantum mechanics: with applications to Schrödinger operators . - American Mathematical Society Bookstore, 2009. - P. 19. - ISBN 0-8218-4660-4 .

Links

Analytical geometry on the plane and in space / Vector algebra (Russian)

Literature

Shilov G.E. Mathematical analysis. Special course. - M .: Nauka, 1961 .-- 436 p.

[_5e1085b5e4a91107-1] Shilov, 1961 , p. 185.

[Blanchard-2] Philippe Blanchard, Erwin Brüning. Proposition 14.1.2 (Fréchet – von Neumann – Jordan) // Mathematical methods in physics: distributions, Hilbert space operators, and variational methods . - Birkhäuser, 2003 .-- P. 192. - ISBN 0817642285 .

[Teschl-3] Gerald Teschl. Theorem 0.19 (Jordan – von Neumann) // Mathematical methods in quantum mechanics: with applications to Schrödinger operators . - American Mathematical Society Bookstore, 2009. - P. 19. - ISBN 0-8218-4660-4 .