Notes On Methods of Proof
As an important result, the inverse function theorem has been given numerous proofs. The proof most commonly seen in textbooks relies on the contraction mapping principle, also known as the Banach fixed point theorem. (This theorem can also be used as the key step in the proof of existence and uniqueness of solutions to ordinary differential equations.) Since this theorem applies in infinite-dimensional (Banach space) settings, it is the tool used in proving the infinite-dimensional version of the inverse function theorem (see "Generalizations", below).
An alternate proof (which works only in finite dimensions) instead uses as the key tool the extreme value theorem for functions on a compact set.
Yet another proof uses Newton's method, which has the advantage of providing an effective version of the theorem. That is, given specific bounds on the derivative of the function, an estimate of the size of the neighborhood on which the function is invertible can be obtained.
Read more about this topic: Inverse Function Theorem
Famous quotes containing the words notes, methods and/or proof:
“The drama critic on your paper said my chablis-tinted hair was like a soft halo over wide set, inviting eyes, and my mouth, my mouth was a lush tunnel through which golden notes came.”
—Samuel Fuller (b. 1911)
“I believe in women; and in their right to their own best possibilities in every department of life. I believe that the methods of dress practiced among women are a marked hindrance to the realization of these possibilities, and should be scorned or persuaded out of society.”
—Elizabeth Stuart Phelps (1844–1911)
“If any proof were needed of the progress of the cause for which I have worked, it is here tonight. The presence on the stage of these college women, and in the audience of all those college girls who will some day be the nation’s greatest strength, will tell their own story to the world.”
—Susan B. Anthony (1820–1906)