In case the parallelogram is a rectangle, the two diagonals are of equal lengths and the statement reduces to the Pythagorean theorem. But in general, the square of the length of neither diagonal is the sum of the squares of the lengths of two sides.
In inner product spaces, the statement of the parallelogram law reduces to the algebraic identity
Most real and complex normed vector spaces do not have inner products, but all normed vector spaces have norms (hence the name), and thus one can evaluate the expressions on both sides of "=" in the identity above. A remarkable fact is that if the identity above holds, then the norm must arise in the usual way from some inner product. Additionally, the inner product generating the norm is unique, as a consequence of the polarization identity; in the real case, it is given by
or, equivalently, by
In the complex case it is given by
Researchers from University of Quebec detail new studies and findings in the area of computational mathematics.
Sep 30, 2008; "Some physical and computational aspects related to the intuitive notion of time in its connection with natural and technological...