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:
“For centuries the death penalty, often accompanied by barbarous refinements, has been trying to hold crime in check; yet crime persists. Why? Because the instincts that are warring in man are not, as the law claims, constant forces in a state of equilibrium.”
—Albert Camus (1913–1960)
“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)
“In any case, raw aggression is thought to be the peculiar province of men, as nurturing is the peculiar province of women.... The psychologist Erik Erikson discovered that, while little girls playing with blocks generally create pleasant interior spaces and attractive entrances, little boys are inclined to pile up the blocks as high as they can and then watch them fall down: “the contemplation of ruins,” Erikson observes, “is a masculine specialty.””
—Joyce Carol Oates (b. 1938)