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:
“In trying to understand the appeal of best-sellers, it is well to remember that whistles can be made sounding certain notes which are clearly audible to dogs and other of the lower animals, though man is incapable of hearing them.”
—Rebecca West (1892–1983)
“I think it is a wise course for laborers to unite to defend their interests.... I think the employer who declines to deal with organized labor and to recognize it as a proper element in the settlement of wage controversies is behind the times.... Of course, when organized labor permits itself to sympathize with violent methods or undue duress, it is not entitled to our sympathy.”
—William Howard Taft (1857–1930)
“When children feel good about themselves, it’s like a snowball rolling downhill. They are continually able to recognize and integrate new proof of their value as they grow and mature.”
—Stephanie Martson (20th century)