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:
“Lap me in soft Lydian airs,
Married to immortal verse,
Such as the meeting soul may pierce
In notes with many a winding bout
Of linked sweetness long drawn out,
With wanton heed and giddy cunning,
The melting voice through mazes running,
Untwisting all the chains that tie
The hidden soul of harmony;”
—John Milton (1608–1674)
“Commerce is unexpectedly confident and serene, alert, adventurous, and unwearied. It is very natural in its methods withal, far more so than many fantastic enterprises and sentimental experiments, and hence its singular success.”
—Henry David Thoreau (1817–1862)
“War is a beastly business, it is true, but one proof we are human is our ability to learn, even from it, how better to exist.”
—M.F.K. Fisher (1908–1992)