Content deleted Content added
TakuyaMurata (talk | contribs) |
TakuyaMurata (talk | contribs) →Methods of proof: section link |
||
Line 69:
Since the fixed point theorem applies in infinite-dimensional (Banach space) settings, this proof generalizes immediately to the infinite-dimensional version of the inverse function theorem<ref>{{Cite web|url=https://r-grande.github.io/Expository/Inverse%20Function%20Theorem.pdf |title=Inverse Function Theorem|last=Jaffe|first=Ethan}}</ref> (see [[Inverse function theorem#Generalizations|Generalizations]] below).
An alternate proof in finite dimensions hinges on the [[extreme value theorem]] for functions on a [[compact set]].<ref name="spivak_manifolds">{{harvnb|Spivak|1965|loc=pages 31–35 }}</ref> (see {{section link||Over_a_real_closed_field}}).
Yet another proof uses [[Newton's method]], which has the advantage of providing an [[effective method|effective version]] of the theorem: bounds on the derivative of the function imply an estimate of the size of the neighborhood on which the function is invertible.<ref name="hubbard_hubbard">{{cite book |first1=John H. |last1=Hubbard |author-link=John H. Hubbard |first2=Barbara Burke |last2=Hubbard|author2-link=Barbara Burke Hubbard |title=Vector Analysis, Linear Algebra, and Differential Forms: A Unified Approach |edition=Matrix |year=2001 }}</ref>
| |||