Pr~f Lie Let 7' Le+ V/t'l'chlef I.e A U - · Proof . OI/,e"rwiSe. ue ~ ase . I. ul . in . fitteR....
Transcript of Pr~f Lie Let 7' Le+ V/t'l'chlef I.e A U - · Proof . OI/,e"rwiSe. ue ~ ase . I. ul . in . fitteR....
(g4-A) Pr~f of a +he/Mwl of $rascamt evnrJ Lie b.
Thegre"., Let Ul be tI. bQZllndM" CcnV'ElK cI~ 7' IR.~. Le+ U be fhe f'Y"s+ V/t'l'chlef e(Jen.fUlttc-litJ" . I.e f-
A l,( = -AI 1{ U >(:),
Th~ ~~ U 11Ulf>b be tJQI. cave .
Pruof We cJ,.~se- tn:f fUnticn UD >0 • ,u')[}Q ==0 sIlc4 -Ma..f - ~ Z{D 's tlnLCa~. S~ch a flvnc-fum a.1lAo/,~ e)( ,. sfr . For excwyI~ / tAle Cevn fr.<,A.e.- the.- em vex Iw. /1 7 +he JrAfh 01 - ,,~ 2(.
{fts/d~ fire (1M<) ~: =AU+),u u/AA=-o
We o.mltme +h a. f 'ZIf -';> 2( • -;J,e f'r s+e(J~"""C-/.b.. We t:rrt j() i~~ -/-t; USe fJe ?r/,ClIXtlnthn. priflcr Ie. !J,f T Je +Ae. bt,jJe,rl-l ?UM1lb~ Sut/, -Ma.+ def ( - V~~~~) ,os clt;f!-M-kr/TliM.s -fh~ Its OAt Xo G H awl tl directz.Qt ~ C{,(,ch IkLJ.
- ~ 'U.)z'l (X__ )-=o
()Ad. f:; 6-M&r- i, f..tf I1.)N (x.) :':-0, Lei ~=-41( I~ 1-Ae 'f}nv(Jtu+~ ()f lO /~
~1 .:: LJ. f(J - I'J C(J I>' - ~,
By.Me ?nI:W~ f't'iltcff1e., 11,',' It :' 0 , ",,~' oS 0
A'fa ~ 0 . TJw.s;
() ~ ~~,. ::: Ll P'il' - 2 Yl CfJ.,./ - 2 'fA.t2..
I { ~ - 2 'fief ~
~ it~~r br CQnv~xi7:' Cflt,' ~ rpfi' Cf/e./e '" (} / 51,' z e;
lite +h~ f {~'f~""" ~ 1'lztwi- j>Yilfeiple.
. Proof OI/,e"rwiSe. ue ~ ase I ul in fitteR. of U.
F'f(Jm '+Ae ~+Ol S inef«,~t-iy
tv lull ~ ('Jul
tvehave.
Lemmo.. 2. U > 0
If ?tof. A-SSUn1e fh~-I- u ~D) =: 0 . Le-/1). (X) =- pJlrX) -r 0 ( X U+E)
. By Hae eruct+{~ Li u. =- ~,u. ~ we iJtUs-f ~ 4fN(X} ==D
5,'helL U (x)~o J I»e AfN.Il ?IV(x) ~o. rlu's cn,:lrd/ctt +~ -fhe /nD-«;~ prihCtf Ie..
Lerwmo.. 3.
~sid1h- -Me floW ~f == L1 Ii -t ).,U > u/~Jl. == 0
Weh~ ...
I...&runa. 1. Fer,.. ~ [}1'Jr-C7)-I/z inif/eJ u" »0, Tieif,...., ex's!:s fl4oo! (]d17'V'8'jes -/-0 fk frr: f eIJev>,r-tai>.·
pynf Le+ Uo =. Z aij;~{!)
LeYJt;Yllt s: (f Yo >0 ,
@ Le~m~ 6 If n ,\S ~. Tk ~ oIL, ?~l9fIJ.)
.--./ I'~ ~x'
prrrof:We solve.. U (x, ... ,. y~ ):: 0 fa ~
th = Xn ('XJ . ",. Aw ~ )
JL ,'s Dm,ve)( '2 pas;+,~
5x~' '>.0 ( ~'t~ Jf"~ lit4h;xJ.
{)SihJ i~llC;; fj~-I;: ~-rhi:Z~ _we1M... ~v An =j;t> TIna $;~
_ U({ -r n,. lA,'" l{;' -+ ?{Jh 211' - 1l,'J{/ Villi so >-0
J Un 'H~ ~~ ~j T/UJ..s lOY' t>rr; (aI''', a,,). If 2. 4.,.l(,. =0 , u.;.e h~
- 11&' at' 9J ~ £. 141 <
FI1Y' ~ a l' p. b, b = rUt, ... It.,) , we Atve.. - u~' (ai tfib,,)(~·tft.0)
+ ...l- £,til' Ih/4f- >0 U
TLr. - l~ Z< is C6'fW€X.