Cap03_Apostol T. M. - Mathematical Analysis

7/28/2019 Cap03_Apostol T. M. - Mathematical Analysis

1/23

M T L G

l r e part o p vious cha e de lt wi a as , ais, t fobj ts. ln s h r ur f r nue of mp ex nu b . d m r ne y, s g di ls

ln hi ar a of u y t c nv e help u met ic in w k o t h e l ,s sp s the e

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t s t c e losed et ncom a t e b e n ng s y f . e s

int sei

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S y, n h ee- im< e s a or r e re ju t s o s e r u f r x ) d e p n -d me ls

D 3 1 Le O / g r. A o d d t f n real u b rs(xh x2, . x ) s l a d m l v cro p

u ally b d c/e e ; fo pr

T numb i. h p t x t k c s l p l

e l - , a s e a o a v di s gs e

m o g A t l t e u spa m s aompl u t on eas r re e . The e de rob

n e t n lysis r ze d g wrh qu i f i a s t ree g s

vect q at on r h n e ca r s h r f sys e gr f .

."


2/23

Anoth r adv nt g i studying -s r a ge l th t w bl tod l in s o ith ma y p es c mmo o 1 , 2 p 3 ce,

. tis, p o e e of me s na y f Hi e -im p ce a s u t n u a i uch re a a d

m . v n -d p q t Alg br a - m n d s ws:

D} 3 2 Lt x = . . . , j ) b nR". W d :

b) Sumx x =

x + y = + . . + M t l c o by (

D e e

Z

, (a r

= +(

r pr


3/23

om(d) u

llx Yl 2 f k) f + +1= Hxl 2 xy l Y 1H i+ 11x llly + l (l xl l ID

Som i e thet l in quality s wri n n zn x -

This ow f om b r p c ngx b d We

llxl t lThe unit c r a v o in " is

is 1 and w o e rema g re ze o. Thu ,

U . . . .lf X U + n 'U

x The v . . . , u ar vc .

L b i n poin R n v n s t v f p ts i R" uc a

f < n f rad u and te B{) or

by B ; .Th b c n f i d ncf m J t a

In p o n r a n 2 i l k,n t l w n a n a u

3 5 Let S b bset of R ,d t t T c d t S p ll n wh po s lo o

t r p n S r u ded b nB( ) n r po fS t i o n n b S. A y co i ng w t r ome m l

D n ion f pn S / its

NO A S i if = S eEx c


4/23

JnR1 of no empty is . T unioof w r al . t tbc u h nd and b a o int al.

Ex mp f s he ; h c r c f w m s al T d o t h a

g u f pla e f t, R2 n co

2 .

e pt e o (Why?) i who! s Eve 1-b lt nR T e rtes n p duc

b1) ( ,

one- men n n er a(a1> hi , . , ( ) i o n c en ope int ral. We d n b , h 1 .

t r t alo n co u

f g ve t

T m Th u n f y c P L F a l ec i f pen e d l un S

S. T e st ong o f se F a x A.Sin o x st l : S B(x S

nd x r n S S e e f S i t

eor m T a j

S A S emp yth g o ov ) x Ak fr 2 nd enc h open L e m l e h iv

x ; S. t n r p

u s h om giv t tsc brb t au o r n s t .Arb a t ct n ,o t

o lw o F e a pl t nt r t n n n va he m / / ) 2 c i t n ne

n o le n jo nt o e

rem k y eno h no e p y p n 1 c o t ne i

dev e

t pr t w i u e nc pt ent


5/23

1 A o e r l d c mp t te f S

S is n n n a/ J n other words a c e in rv lof S is a p rsubs of yo h in a co ta

T/rlm 3 10 Ev y f a m belo s t ne n no p in rval f

Pr j A u e S. T i d n me n The ear m s ch t s bu h" l gest of h d ec -po h r verify t hi

(x) nf { (a, x) S}, b(x) up {b: (x b S}.H (x) g - d x) ight Cle r y,th e is n n v l

u h o 1, i compn nt in val n i ng f S c ai gx, e i n J" s n

n n n n g I" a , Henc do p l ow ha J" ! J o

/; J,.heo em (R pr s n io t mfo p n 1e a ) E y

e u f b/ a

P o f S h co v S gxof h n lx tw of and o n i

, h r n n i t nS d o nb ( H d / , m i j n o o

It h th h l n.o pu p ,{x1, 2 3 } d n t b o at umb . ch o

l x inf n ely ma 0 but h r x y w t W th de n io Fy f q ti F(/ ) i a o wi h l tind

F F p I" d h v x. n nd h p e Th o e o

d b w th t l n u .h i

T h p e nt ti no q n c ; S j u d j t h h s v u h c p nen S. a

m d quo

i v lth n he p fo n on om nv l nar y -o r a R1 b


6/23

w

the uni of w nonemp y sjo o n .This r t bg t a a o terv l h c n o ec d sf

n R" b i uss t Se 4 16

SE set R is ed os !f icompfe e

is o n.

E [ J n R' ] ]

o dime s al c v x , q n eof Th 3 7and h w t

fr r lo e t o g

u a t e f e ,d hr e fan rb t y c / l e i o e .

A l o nd d s d d t o

f pen a B l

P W n h A n (R h n t o o nd B R" ) t inte c d

AH RENT C o d ca b d n f dh r t an n

J x p R" y; T ex hto S v r -b B(:)

o

Ifx E S r h r - Ii u f R w b , t p : o

e po n s a h y b B s om e c m .

D R i c l edum fS - ea r e x


7/23

ln ot er wo ds, i an um l opoi f if, and y ad s tS {x}. If sno c umu a i p of . the n

solted S

e o the o 1{ , . , h O an acmul tion pint. n mb .Ev in of [ b] c t f he

t op is a ac m ul t o p f S, hen

y a fS.

Pr A he c r ry; pp 1-b ( ) x which c tonly e r t c m x, Ith tf h o v br

-b a u wh o o s d n tom . T .

T m pl r a h a

u n t c n s o w t Te w

v i g F e m i g { um o n n h tha o ai w h v an u a

k a z o-We a

J.7 SED SE lo w o p m f h

r w .

A ly a .

P d d o W S We b t R iR o , b R u B i oS g c

T c l i a r ash - S o dh e tHen o b l B(:) i t S R S. Th r R" S

a d eS

9 o n set S i d f S b


8/23

ny haveS s c pointof r heo m lsh w t t p osi S ds n only

:

S of al ccum / ti n p sof a c / d he erive o n y

Cl n impli dos and o ly ln t r s, h v

R c d only c a a c mul

b d A b

- r) fo mer B ZW rst a . Jf o e R n t

h Rw nPr f To i w gv t o f S n is bunded,it e v a] A n thub v l O]

n n it u b rv l[ . 1 1b a bint er al [ 2]con ining ni t b t n c nuth s p ss nwa col ti f nt rvals the hine v l [ in l h Cle th p

n ponts h in r ge d ,mus q , ay [Wh q l? h c m on p n ,n v h v [ , bJ B(x; s on s

ar gh h w a t t m n h a u l on p t th i b ong

p r R" n x n of a in t e p l t r f n R by

F g.in b un e ,S e o e 8 0 a a he- m s a v /1nby e q it

a S H 1 not pr c

(k

J i = If1) x 1l x ;n h t po , x , " / > n c i

on -d ension t r al E chi teral P n b e


9/23

-

:>1

_ r

- l-l r 7 _ l F ig 3 . 1

rm t subinterv l I d by th in qu l t s

Ne wc d r al p l sian du h ( )

w h k; . x l u p d, of , cu pr s - m o t . T uof th s v l t

t 1 h S; e l tu 1 m o nt of S.O f by te xp

! /2 t b val o W o eh J d d J t g t1p> a ar v g an - m a u t SIf t h oce ,

t u a l l ,th t th pr ha itc t n n u nd

p h

v

2, t p f a p t "\m 2 mu t

al th gh e s m

d h ru ot y W t t h t 2


10/23

c l io ntof T T t, o fngst ea h o he i a J1, 2

m su h / "-2 rj , n g ll . But c Jontai t m ypi s o S w B ; r), h p s e a cu ul t n S

CR ITEON ORA pp f z W st s o mp he Ctor

e o m

t {Q Q2

... . .

Q is ed Q1 ! b .Th t r c on Pr Lt S =i d u T

o t S. W n- p e p f N w co o i s = Q S c e

y J: ' w e S b v h

is c um h Qb r c Bu vn g ho mp t f A. d x{ b ) e bl g Q ig h

co y o Q1 T umQ a d 1

l e c pt o c ng o t d ohL f Te

F, F all S

1 The o i n rm = 2, 3, . .. ), s aopn c e g t a f a u b i . Q T s r-v l r (


11/23

O d d O snot

n . of kw tion . 8

T e Li d lf v r ng t ev o n rin of a t Sa s c u b T

h ow g p m a r u :

{A A.h . } d note h c un abl ba h ing a d rs p w n

R8 S op R x. T then-b x d n , av

x T o n co b T m 2. 7

t n n 1- a B(x; } S. W U ndap wi h r n l co h s " r x nd, r. l d i s h )

n x

l r l r I f eahk . T

xH IY1 - X I - x I 4N xt r i n num r r/4 r/2 n B ; q) dB(y q : q n eore v

3 r

s ; U h u b /l i v

} c c n o -bl r i n T s h

b l n F ch


12/23

Assume T n e is an n t Fsuch t By a nba ; t S h e u ei ly

m uc ; di g o a S, but ch ly f r x-p e,th

d x s y m(x ). e

e < S.

e - ll a x v v m s o bl i n of hi h sA. To s fF .

wh w plyco e ac on S f F wha J s m th pr

1 L T L d s a c e b aw c o b v gT e H B u t we k w w a d

a o g e pr f the C r

& F ln R". T /F

A u e F, s { 1 /2 } A b T h

T h u p W h l v e o h u

F p e s h e R w h dDe u o {Q > 1 wQ nd fo m >

Q A S ).Th t Q s fth sep h h u Iwe h w

u m Q e e h r o A d S w d

O r t pr t s s : Ech Q

i h o e h c R Th Q g S c t Q .

g u d T t e p yh n h o

. Th e t t wh h sQ wha t m h g u he ts S u h s m bin

;

=S" T e o e Q e h


13/23

S RWe have just e tha set in R o a d ound , e ny opc veri g of S r e uce t ni cov i g lt is na u allo i q whether

h

igh e o er d u e s t w o ty. u s rS in R" s a be compa t nd nly

r v ing fS te ubcov r, h ub o ecti n S

T -Bo l t tha v y eand b d m a t. Now w p ov e ul

L t S h s f . T m aq al :

) S pa .) S s d a b .) E y S has a a n S.

n a ve, (b) impf ( lfwe p ovet ( ) im l hb)mpl ( (b), s bl h v l cof a h e

As ume (a) ho d W r s bou d d. C o o pThe \l nf nb l {p; k , 1 o ri g om ct e s l al S c S s bo .

Nex rove h S . p s Sis d T t sumul t o S c S lfx l ' l E

i e San co c { B(x;: S} o n c v jg

S B a ni n b r o t ig b r ood cov r ,

t o t r i 1 h . . p e i i y o ro t the B y; i m a y b x1 "ln

r n ll rk, n b g i u i y - k o

l lll 11 ll l 2r ll llke S i co tr dict g t t

c u u oi T i n S ani i b

( ) h ds t e p oof f d ,b T n t f ou ( n S bou nd by

B z -W a t o e as ac a t x ay N w a


14/23

an um l tio int of S a d h x S s c is l d The r )imp

h l W v is u u d , r v a x, { 1 , x2 .}

n ub t of a , by (e) T s um ion B > w

- i > m Uc nt d i g c mu t p t his r s t S

nde .T c mple he pr ofw m s ow L ! um a

o S Si vry b h o x t l ns d h r o s B(x; /k b a d i t .y T {x x a i S btha

: al ul n t c b e S, e f the u th t n umunt r w b r e s wth yati p n

T d up t t x Th y g l

Y - x : - < - fk E TI k s ge th t fk 0 i a

d < - x h h B(y; ) nk 0 i Hen c u lat o pl

p i b .

e p of o ap ly f w r an n nt n th t R"

i f abtr the l d h c n p f a & A a no mp y s t of bj ts

(cl p :) O her w r ric /c t g h f ll w f u f

O

y d y x4 + y

n gt v is t ron h n t u v g h ugh P

4 l r q .


15/23

We so e mes eno e a metric s by {M, d) t mpha iz bo e d the d ni of

M R:; d(x y) l t W r K w l he t E lidno c s s ly

e. t mp l ; d(21 Z ) 2 - As g le f mE l n R2 e d lhe

3 M no y (x O = d x y}= 1 ir # y T l a d M d a d crte r

d

a y no mp y u of M,

is

w h e mor r nofW l t r rn bS e f F r x pJ he n l m s Q - YI su

M R d x, .(x1 } )2-+ 2 wh = 2 T m (M, is not a E Rbc u m r id e .

6 M

{ J u x,) g of

l join ng i c r .7 M = h

a ng j g p n M R d !

9. R d I xIN S TO L GY MC PA

Te b ot f n t b :d o b M

I E B ; r) w u t of M u h t a

d(x, a) < r.

S e n bby B1 a; p c i

s a b p o Bsa s l int h h B ( ) l Eucl a sp R O1)

S O1 1) s e R "sp er c t

E m tr .(S Ex e M, a a s S M a

inter o S A i


16/23

m

c edo n M a i point a t r c closed M fs

1 r a M f f

of of S n. eu o

l su [ o 1 Rh w e ub f M n t

ext r m io w M S (S d) be a m tric pa of b t of

S T X i n n d ly X = A

A wh h p M

Pr A e n M X n S. I s m > OH eB5( ; ) S A S X .ve y, m X s S W S r s F r in X B x; n n nX. Now

B J S U ": J

A M A = 3 3 L(S, d) s s (M, t f

S. f B wh o

P S B n n AS n S S

C r n t - T X = n S - ( n S S (M A) S B

B - c1 d nM p M p f r B x;

c t l ne of I - } au u a o S f a d h v t ' t mu a t S.


17/23

e owing th m val dn v r r d and pxac y E the r f , th dis

Ux - y ee onl pla by m r c

T e union / a y co e t o/ p sei s p ,a d th n r-s cl o /a j c li n f p n p .b) T un j c J t f d d at s c /

e ti n f / A is B d, h - B B A

3 F r n b S f w ngsta em n ar q le :d b e .

) a n ac u ul o i .d) S

= Q f Q.

O r B z W r r : h r m, C or a veri g r f d H -B e! u h

p eo Eu "b l o R n g an b r c M,d). Fu er r s c on:M

eq d x n s me . O f te i ] in 3

i b s ar c

5 L M,) p J b a u f F o n

M b en ver go U A.S M alJ f ve n c n

S c 1 u S B( ; r r m > M. L S m a u fm r p e M Th n:

) S d .)Ev ry n n fS h a mu n S.

P To ( r d r hgu tw h w a p T y t e E c

Hx gh uby e tr c x


18/23

p ii gue y c nt d o ni a dtha oint of i a mul t o i f ac in

sa al B( ) n a n npo T c n

f f x As br g nion f e ao i mp , te su o o v Sa d hv T B t c t i b u T d ach l

o t i m nt of

n E c id " p rti s iand ( ) seq ival o m-31) g n l r p r ii se u

c mp t ess a Re e 3;4 u so 3 42

g ves xam l o m Mwb ch n ot

3 j X be a ub t/ M. T

P L F b a n ri f y UA wi sh w an u b o e t o e Sin X s o di o e e M -

, o { M - } s c g M i mn n a na n v w m d M X

l " V M X v d M - X o x. e ca

e t mth o d il v r X T A"s

D E 3 0Lt a e ofa x

b u p t B x; r Cn i s l e fS p t M T t a mp t c h y

o d

f tS -S

Thi r a s ows t o n

E B r - = r um h Fur r pro s a the Exer s a d al ip e


19/23

R1 a o R 1 is t tha cl nt o

: m in s of lowing s R d d i e wh hh r ( b ).

t .

f

d) ume) A u

g u o e / Q/m / /

.

) . .)

. =

J J e E 3 r e w g R2:a A zsuc a z

z uc z !e) p ex ld}

A i s X

3.4 PoV n S R s t lumb3 Pv e y R c R1 a m lr ?

R t o of co n o n

7 h u i R is h r c e tobta o i g i c t

op d

R".8 Po h t a d - o t Js o

3.9 Pov i io o o R"0 I R , n of l o r co tai

. s by y th s r p u f T f

( T t T, ( S ( T


20/23

L t S' h t the u of a t :n R'; h

th n ' v

= S' e) i n R Si i nof a s b R" ai g T i tal o n g

1 of R". e nd .

T T e n n 9 u a e e n any m c t i for v p r of an nS nd r al

at g + J t -lly a d R 3) d ) Ev y is o v x

b E m n o l i te a c- v x.

of con x t x. c of a o t .

Lt i f and l t w g a m n ,eith g p f r unt pl .

um t o f the x mu1at ch

) I a u t o po n f S, h x s u l o f e o i

16 o l um a (0 c t e p e oun a l c t H . W 2 }

a e k p, q {Q.} der l a Q0 Q uch t t , n th C r e -

si n a nt t o

R , v e t a l tnt f b8P t t d - w h r ) n

> n , nt b g } o n f in s f {l/ 2 ) = . o c v g f h a (O . P ve u gT o

o ti v (0,G a o a c s d xhib t n a e

p v r g t G w th r e x B(x

cb t B n a l ov ha abP h c l o sj y b G

ampl a f sjo cl n


21/23

67

Assu e th in id to f C bB(x e o ty th B ) is not o b . t is nc a l , t o nt of

A ha s h i no n ble. L T e e ff ;

a - un le) T is ,

S is l c u b e) T 1 s.

N e Exer 3 2 i i f

R" l d i i f cl wh i P v ha eve u bl o R xp si

= A vB w s f t u ble {CH

l y m r p v 0 h w ad clo

3 Z7 C r g m i

d ) = m a , - y ,I d2(x , y=

L lx1- Y Ii C ;I h g prov b l .(a; r)

:R2 1 S t a o

b (2 uaJ p l l

d A R3, d.J d 27 l l - Yl uE id e t f l al t d n ":

d ll l ; l Y If M, d

' = d' a m r c fr ' < f , i

n su a sp n m pa e (M he c f r u >O u

= {x } Pov ; r i cl G m h t s f

B


22/23

M,if u s sati fyAs he l of th oA id i S. mpl Q of na de R A s d n S S n T pr vtha n T

3 Re r o

M t a / f

h e sksub t A wh is in M. F r pl , Q alm a cun ble v t Euc n b e

R P that Lnd f v g svi n y l

xe 3.32 A a d B i S p vethat B

to E 3 2 U dB S dB i S p B s e SG vn tw (1 ) S2), C n pr .t

1 c ru i exa l x ( 2)a y = ( 1 + 2 x y2 .P t p

me c f rS 2 fw x p

COl 1 w g b rK

s M d ly i m t

I S T b M

4 u C Q m h

f a n m n l b a S a Q

d B a M :

in A = M Min (M A in A

3 a) i 1 : ( nt A b) i t< " , A F i t co fe G v p w

J 7 EF As cuAb G v x nit c k F d h d (a)

3 4 n A0 Gi t 8A


23/23

3.4 If int A B M, e (u B) 0G v le wh = B= b u B) =A d A .

3 Ir ( 0 tb A VB

Y

PA . M ph No 13 W N12 l

M - Al , 9 G M H

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Cap03_Apostol T. M. - Mathematical Analysis