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:
“At the end of one millennium and nine centuries of Christianity, it remains an unshakable assumption of the law in all Christian countries and of the moral judgment of Christians everywhere that if a man and a woman, entering a room together, close the door behind them, the man will come out sadder and the woman wiser.”
—H.L. (Henry Lewis)
“Man’s main task in life is to give birth to himself, to become what he potentially is. The most important product of his effort is his own personality.”
—Erich Fromm (1900–1980)
“Le silence éternel de ces espaces infinis m’effraie. The eternal silence of these infinite spaces frightens me.”
—Blaise Pascal (1623–1662)