Pr~f Lie Let 7' Le+ V/t'l'chlef I.e A U -  · Proof . OI/,e"rwiSe. ue ~ ase . I. ul . in . fitteR....

4
(g4-A) of a +he/Mwl of $rascamt evnrJ Lie b. Thegre"., Let Ul be tI. bQZllndM" CcnV'ElK 7' Le+ U be fhe f'Y"s+ V/t'l'chlef e(Jen.fUlttc-litJ" . I.e f- A l,( = -AI 1{ U >(:), T U 11Ulf>b be tJQI. cave . Pruof We tn:f fUnticn UD >0 ,u')[}Q ==0 sIlc4 -Ma..f - Z{D 's a flvnc-fum a.1 e)( ,. s fr . For excwy / tAle Cevn fr.<,A.e.- the.- em vex Iw. /1 7 +he JrAfh 01 - 2(. fire (1M<) =AU+),u u/AA=-o We o.mltm e +h a. f 'ZIf -';> 2( -;J,e f'r s + e (J We t:rrt j() -/-t; USe fJe ?r/,ClIXtlnthn. priflcr Ie. !J,f T Je +Ae. bt,jJe,rl-l Sut/, -Ma.+ def (- ,os clt;f!-M-kr/ TliM.s Its OAt Xo G H awl tl directz.Qt C{,(,ch IkLJ. - 'U.)z'l (X__ )-=o ()Ad. f:; 6-M&r- i, f..t f I1.)N (x.) :':-0, Lei 1-Ae ()f lO .:: LJ. f(J - I'J C(J I>' - By.Me 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 b r Cflt,' rpfi' Cf/e./e '" (} / 51,' z e; lite f 1'lztwi- j>Yilfeiple.

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....

Page 1: Pr~f Lie Let 7' Le+ V/t'l'chlef I.e A U -  · Proof . OI/,e"rwiSe. ue ~ ase . I. ul . in . fitteR. of . U. F'f(Jm '+Ae ~+Ol. S . tv . inef«,~t-iy. lull ~ ('Jul . tvehave. Lemmo..

(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.

Page 2: Pr~f Lie Let 7' Le+ V/t'l'chlef I.e A U -  · Proof . OI/,e"rwiSe. ue ~ ase . I. ul . in . fitteR. of . U. F'f(Jm '+Ae ~+Ol. S . tv . inef«,~t-iy. lull ~ ('Jul . tvehave. Lemmo..

. 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..

Page 3: Pr~f Lie Let 7' Le+ V/t'l'chlef I.e A U -  · Proof . OI/,e"rwiSe. ue ~ ase . I. ul . in . fitteR. of . U. F'f(Jm '+Ae ~+Ol. S . tv . inef«,~t-iy. lull ~ ('Jul . tvehave. Lemmo..

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 ,

Page 4: Pr~f Lie Let 7' Le+ V/t'l'chlef I.e A U -  · Proof . OI/,e"rwiSe. ue ~ ase . I. ul . in . fitteR. of . U. F'f(Jm '+Ae ~+Ol. S . tv . inef«,~t-iy. lull ~ ('Jul . tvehave. Lemmo..

@ 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.