The Parallelogram Law in Inner Product Spaces
In a normed space, the statement of the parallelogram law is an equation relating norms:
In an inner product space, the norm is determined using the inner product:
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:
Adding these two expressions:
as required.
If x is orthogonal to y, then and the above equation for the norm of a sum becomes:
which is Pythagoras' theorem.
Read more about this topic: Parallelogram Law
Famous quotes containing the words law, product and/or spaces:
“All men, in the abstract, are just and good; what hinders them, in the particular, is, the momentary predominance of the finite and individual over the general truth. The condition of our incarnation in a private self, seems to be, a perpetual tendency to prefer the private law, to obey the private impulse, to the exclusion of the law of the universal being.”
—Ralph Waldo Emerson (1803–1882)
“Poetry is the only life got, the only work done, the only pure product and free labor of man, performed only when he has put all the world under his feet, and conquered the last of his foes.”
—Henry David Thoreau (1817–1862)
“Surely, we are provided with senses as well fitted to penetrate the spaces of the real, the substantial, the eternal, as these outward are to penetrate the material universe. Veias, Menu, Zoroaster, Socrates, Christ, Shakespeare, Swedenborg,—these are some of our astronomers.”
—Henry David Thoreau (1817–1862)