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by: Reyes Glover

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# COMPLEX GEOMETRY M 392C

Reyes Glover
UT
GPA 3.67

Daniel Freed

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COURSE
PROF.
Daniel Freed
TYPE
Class Notes
PAGES
2
WORDS
KARMA
25 ?

## Popular in Mathematics (M)

This 2 page Class Notes was uploaded by Reyes Glover on Sunday September 6, 2015. The Class Notes belongs to M 392C at University of Texas at Austin taught by Daniel Freed in Fall. Since its upload, it has received 53 views. For similar materials see /class/181432/m-392c-university-of-texas-at-austin in Mathematics (M) at University of Texas at Austin.

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Date Created: 09/06/15
10 APPENDIX Technical results 101 The inverse function theorem Lemma 101 Contraction mapping principle Let M be a complete metric space and suppose T M a M is a map such that dT96Ti mm where h lt 1 Then T has a unique cced point Proof Choose any point zo then dTmz0Tnx0 S hmdwo7 Tnimpo for n 2 m kmdz0 T950 dTx0 T2950 dT m 1z0T mz0 km1 k kn m 1dz0 T950 km S d07T0 This is a Cauchy sequence7 so completeness of M implies that it converges to x Thus z lim T zo and so by continuity of T7 Ta lim Tn zo z For uniqueness7 if Ta z and Ty y then Elmo dT96Ty S kd9 y and so h lt 1 implies dy 0 D Theorem 102 Inverse function theorem Let U Q R be an open set and f U a R a 0 function such that Dfa is invertible at a E U Then there eccist neighbourhoods VW ofa and fa respectively such that fV W and f has a C inverse on W Proof By an af ne transformation z gt gt Ax b we can assume that a 0 and Dfa I Now consider gm z 7 By construction D90 0 so by continuity there exists r gt 0 such that if lt 2r7 1 Dm lt7 Hall 2 84

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