+QMiBMmBiö /ǰmM2 7QM+iBQM p2+iQ`B2HH2

+QMiBMmBiö /ǰmM2 7QM+iBQM p2+iQ`B2HH2

.Mb iQmi H2 +?TBi`2E 2iF bQMi /2b2bT+2b p2+iQ`B2HbMQ`KûbX

G2 #mi 2bi /QM+ /2 ;ûMû`HBb2` H2b MQiBQMb /2 HBKBi2b 2i +QMiBMmBiû ¨ /2b 7QM+iBQMb ¨ p`B#H2b 2ifQm pH2m`b p2+iQ`B2HH2bX 1M T`iB[m2- HǶûim/2 bǶTTHB[m2` ,

ě mt 7QM+iBQMb MmKû`B[m2b /ǶmM2 Qm THmbB2m`b p`B#H2b `û2HH2b c ě mt 7QM+iBQMb /ǶmM2 p`B#H2 +QKTH2t2 Uz7!1/z-z7!e^z ĘV

ě mt TTHB+iBQMb /ǶmM2 p`B#H2 Ki`B+B2HH2 UM 7!det(M)-M7!M ¹- ĘV ě mt TTHB+iBQMb HBMûB`2b Qm KmHiBHBMûB`2bXXX

A ě GBKBi2 /ǶmM2 7QM+iBQM p2+iQ`B2HH2 RV .û}MBiBQMb 2i +`+iû`BbiBQM bû[m2MiB2HH2

aQBi f ,D⇢E!F- a2D 2i`2FX PM bmTTQb2DMQM #Q`Mû /Mb H2b /û}MBiBQMb p2+ kxk ! 1X

URV , f(x)_x!a!` () 8">0- 9 >0 i[8x2D- UdE(x, a)6 )dF(f(x),`)6"VX

UkV , kf(x)k !_x!a1 () 8M2R-9 >0 i[8x2D,(dE(x, a)6 ) kf(x)kF >M VX

UjV , f(x) !

kxk!1` () 8">0- 9M 2Ri[8x2D- UkxkE>M )dF(f(x),`)6"VX

U9V , kf(x)k !

kxk!11 () 8M2R-9N2R i[8x2D- UkxkE>N ) kf(x)kF >M VX

U8V , .û}MBiBQM ;ûMû`B[m2 ,

TQm` iQmi pQBbBM;2V /2`- HǶBK;2 `û+BT`Q[m2 f¹(V) ={x2E|f(x)2V}2bi mM pQBbBM;2 `2HiB7 /2aX

UeV , *b T`iB+mHB2` , 2ti2MbBQM mt HBKBi2 ¨ /`QBi2 2ifQm HBKBi2 ¨ ;m+?2 [mM/D⇢RX .û}MBiBQMb , HBKBi2 /ǶmM2 7QM+iBQM p2+iQ`B2HH2 HQ`b[m2 xi2M/ p2`ba, URV

URV , f(x)_x!a!` ()TQm` iQmi2 bmBi2(xn)2D^Ni2HH2 [m2xn_n!1!a- QM f(xn)_n!1!`X

UkV , f(x)_x!a/!` () 9">02i(xn)2D<sup>N</sup>i2HH2 [m2xn<sub>n!1</sub>!a 2id(f(xn),`)>"X

UjV , kf(x)k /!

x!a1 () 9(xn)2D^Ni2HH2 [m2xn !

n!1a 2i(f(xn))2bi #Q`Mû2X h?ûQ`ĕK2 , +`+iû`BbiBQM bû[m2MiB2HH2 UkV

kV S`QT`Bûiûb

URV , aBf /K2i mM2 HBKBi2- HQ`b +2ii2 HBKBi2 2bimMB[m2X PM H MQi2lim_x!af(x) = limafX

UkV , aBa2D 2if(x)_x!a!`HQ`b`=f(a), QM /Bi [m2f 2bi +QMiBMm2 2MaX

UjV , aBf mM2 HBKBi2 }MB2 2M aHQ`b BH 2tBbi2 mM pQBbBM;2 `2HiB7 /2abm` H2[m2Hf 2bi #Q`Mû2X S`QT`Bûiûb UjV

URV , G HBKBi2 /ǶmM2 +QK#BMBbQM HBMûB`2 2bi H +QK#BMBbQM HBMûB`2 /2b HBKBi2bX

UkV , aB(F,+,⇥, .)2bi mM2 H;ĕ#`2 MQ`Kû2- H HBKBi2 /ǶmM T`Q/mBi 2bi H2 T`Q/mBi /2b HBKBi2b-

UjV , GBKBi2 /ǶmM2 +QKTQbû2 ,bQBi f:D!F-g:D⁰!Gp2+f(D)⇢D⁰X aQBia2DX aBf(x) !

x!ab2ig(y) !

y!b`- HQ`bg f(x) !

x!a`

U9V , 1M T`iB+mHB2`- bBf(x)_x!a!`- HQ`bkf(x)k !_x!ak`kX PTû`iBQMb bm` H2b HBKBi2b U9V

R

URV , G MQiBQM /2 HBKBi2 2bi BM+?M;û2 bB QM `2KTH+2 H2b MQ`K2b /2E 2iF T` /2b MQ`K2b û[mBpH2Mi2bX

UkV , GQ`b[m2E2iF bQMi /2 /BK2MbBQMb }MB2b- +2ii2 MQiBQM 2bi BMi`BMbĕ[m2X LQ`K2b û[mBpH2Mi2b U8V

úim/B2` H2b HBKBi2b 2M(0,0)/2b 7QM+iBQMb bmBpMi2b ,

URV , f₁(x, y) = x²y x²+y²

UkV , f2(x, y) =_x2^xy+y²

ii2MiBQM ,TQm` ûim/B2`lim(x,y)!(0,0)f(x, y)- BH L1 al66Ah Sa /Ƕûim/B2`lim_x!0lim_y!0f(x, y)XXX 1t2KTH2b UeV

aBF 2bi /2 /BK2MbBQM }MB2- 2M MQiMif(x) = (f₁(x), f₂(x), . . . f_n(x))/Mb mM2 #b2(e₁, e₂. . . , e_n)/2 F- BH v û[mBpH2M+2 2Mi`2 ,

URV , H 7QM+iBQM p2+iQ`B2HH2f +QMp2`;2 2Ma

UkV , H2b 7QM+iBQMb +QQ`/QMMû2bfi:E!K+QMp2`;2Mi 2Ma 1i /Mb +2 +b- lim

x!af(x) =⇣

x!alimf1(a),lim

x!af2(a). . .lim

x!afn(a)⌘

= Xn

i=1

x!alimfi(a)·ei

1M T`iB+mHB2`- mM2 7QM+iBQM ¨ pH2m`b +QKTH2t2bf:D!C+QMp2`;2 2Ma2D ,_2(f) 2i AK(f)+QMp2`;2Mi 2M a 2i HQ`blim

a (f) = lim

a (_2f) + lim

a (AKf).

TTHB+iBQMb +QQ`/QMMû2b 2M /BK2MbBQM }MB2 UdV

aBF =F1⇥F2⇥. . . Fn- 2M MQiMif(x) = (f1(x), f2(x), . . . fn(x))p2+fi(x)2Fi- BH v û[mBpH2M+2 2Mi`2 ,

URV , H 7QM+iBQM p2+iQ`B2HH2f +QMp2`;2 2Ma

UkV , H2b 7QM+iBQMb p2+iQ`B2HH2bfi:E!Fi+QMp2`;2Mi 2M a 1i /Mb +2 +b- lim

x!af(x) =⇣

x!alimf1(a),lim

x!af2(a). . .lim

x!afn(a)⌘ . :ûMû`HBbiBQM m +bF 2bi mM 2bT+2 T`Q/mBi , U3V

jV *QKT`BbQMb /2 7QM+iBQMb

aQB2Mi f 2ig /2b 7QM+iBQMb p2+iQ`B2HH2b 2i mM2 7QM+iBQM `û2HH2X PM /Bi [m2 f 2bi Mû;HB;2#H2 /2pMi m pQBbBM;2 /2aHQ`b[m2 ,

URV , f(x) =o( (x)) () 8">0 -9 >0i2H [m28x2D-d(x, a)6 ) kf(x)k6 (x)U TQbBiBp2VX PM /Bi [m2 f 2bi /QKBMû2 /2pMi m pQBbBM;2 /2aHQ`b[m2 ,

UkV , f(x) =O( (x))() 9M 2R-9 >0i2H [m28x2D-d(x, a)6 ) kf(x)k6M (x) U TQbBiBp2VX PM /Bi [m2 f 2igbQMi û[mBpH2Mi2b m pQBbBM;2 /2aHQ`b[m2 ,

UjV , f(x)⇠g(x) ()f(x) g(x) =o(kg(x)k) U+Ƕ2bi mM2 `2HiBQM /Ƕû[mBpH2M+2VX

*QKT`BbQM bvKTiQiB[m2 UNV

aB 2bi ¨ pH2m`b bi`B+i2K2Mi TQbBiBp2b ,

URV , f(x) =o( (x))()f(x)/ (x)_x!a!0X

UkV , f(x) =O( (x))()(f(x)/ (x))2bi #Q`Mû2X

UjV , f(x)⇠ (x)()f(x)/ (x)_n!1!1X

.Mb +2 /2`MB2` +b /2 HǶû[mBpH2M+2- f 2bi `û2HH2 2i f(x)>0TQm` iQmixT`Q+?2 /2aX

U9V , f(x)_x!a!`()f(x) `=o(1)X

U8V , 1M T`iB+mHB2`-f(x)_x!a!0()f(x) =o(1)X

UeV , f 2bi #Q`Mû2 m pQBbBM;2 /2a()f(x) =O(1)X

*`+iû`BbiBQMb f S`QT`Bûiûb URyV

GǶû[mBpH2M+2 2Mi`2 7QM+iBQMb ¨ pH2m`b `û2HH2b 2bi +QKTiB#H2 p2+ H KmHiBTHB+iBQM- H /BpBbBQM 2i HǶûHûpiBQM ¨ mM2 TmBbbM+2 +QMbiMi2X SQm` iQmi2 mi`2 QTû`iBQM 2Mi`2 7QM+iBQMb û[mBpH2Mi2b- û+`B`2f(x) =g(x) +o(kg(x)k)- bm#biBim2` 2i bBKTHB}2`X

h?ûQ`ĕK2fJûi?Q/2 URRV

k

URV , .ûi2`KBM2` H2b HBKBi2b bmBpMi2b- HQ`b[m2 +2HH2b@+B 2tBbi2Mi , RX limx!0

p1+x p 1 x x

kX lim_x!+1ln^x_x+x^px

jX lim_x!0+x^x

9X lim_x!1+lnx.ln(lnx)

8X lim_x!0(1 +x)^1/x eX lim_x!1arccos^{1 x}_x

UkV , JQMi`2` [m2 iQmi2 7QM+iBQMf:R!RTû`BQ/B[m2 +QMp2`;2Mi 2M+12bi +QMbiMi2X

1t2`+B+2 R , _TT2Hb /2 i2+?MB[m2b /2 JSaA ,

AA ě *QMiBMmBiû 2i +QMiBMmBiû mMB7Q`K2X RV *QMiBMmBiû

aQBi f mM2 7QM+iBQM /û}MB2 2MaX

URV , .û}MBiBQM HQ+H2 ,f 2bi +QMiBMm2 2Ma()f(x)_x!a!f(a)()f mM2 HBKBi2 2Ma U/Mb +2 +b- +2ii2 HBKBi2 M2 T2mi āi`2 [m2f(a)XXXVX

UkV , .û}MBiBQM ;HQ#H2 ,f 2bi +QMiBMm2 bm`D() 8a2D-f(x)_x!a!f(a)XLQiiBQM , C(D, F)X .û}MBiBQMb URkV

f:D!F 2bi +QMiBMm2 2M a2D, 8(xn)_N2D^N, Å

xn_n!+1! a)f(xn)_n!+1! f(a)

*`+iû`BbiBQM bû[m2MiB2HH2 /2 H +QMiBMmBiû HQ+H2 2Ma, URjV ã

URV , G +QK#BMBbQM HBMûB`2 /2 7QM+iBQMb +QMiBMm2b 2bi mM2 7QM+iBQM +QMiBMm2X

UkV , G +QKTQbû2 /ǶmM2 7QM+iBQM +QMiBMm2 2bi mM2 7QM+iBQM +QMiBMm2X

UjV , aBF 2bi mM2 H;ĕ#`2-H2 T`Q/mBi /2 7QM+iBQMb +QMiBMm2b 2bi mM2 7QM+iBQM +QMiBMm2X

U9V , aB F 2bi mM +Q`Tb- H2 [mQiB2Mi /2 /2mt 7QM+iBQMb +QMiBMm2b /QMi H2 /ûMQKBMi2m` M2 bǶMMmH2 Tb 2bi mM2 7QM+iBQM +QMiBMm2X

U8V , *QMiBMmBiû /2b TTHB+iBQMb +QQ`/QMMû2b ,

aB F 2bi /2 /BK2MbBQM }MB2- HQ`bf :D!F 2bi +QMiBMm2 bB 2i b2mH2K2Mi bB iQmi2b b2b 7QM+iBQMb +QQ`/QMMû2b /Mb mM2 #b2 /2F H2 bQMiX

UeV , :ûMû`HBbiBQM , aBF =F1⇥F2⇥. . .⇥Fn 2bi mM 2bT+2 T`Q/mBi- HQ`b f :D!F 2bi +QMiBMm2 bB 2i b2mH2K2Mi bB iQmi2b b2b 7QM+iBQMb +QQ`/QMMû2b ¨ pH2m`b /Mb +?[m2FiH2 bQMiX

S`QT`Bûiûb UR9V

SQm`E [m2H+QM[m2 ,

ě H2b 7QM+iBQMb +QMbiMi2b bQMi +QMiBMm2bX ě H 7QM+iBQM IdE : (E,k kE) ! (E,k kE) 2bi

+QMiBMm2X

ě H 7QM+iBQM IdE : (E, N1) !(E, N2) 2bi +QMiB@

Mm2,N1 2bi mM2 MQ`K2 THmb }M2 [m2N2X ě H 7QM+iBQMk kE: (E,k kE)!(R,|.|)2bi +QMiB@

ě iQmi2 7QM+iBQM HBTb+?BixB2MM2 2bi +QMiBMm2XMm2X ě H 7QM+iBQMd_A:x7!d(x, A)2bi +QMiBMm2X

SQm`E /2 /BK2MbBQM }MB2 , /Mb iQmi2 #b2 , ě H2b 7QM+iBQMb +QQ`/QMMû2b bQMi +QMiBMm2b- ě iQmi2 7QM+iBQM TQHvMQKBH2 2M H2b +QQ`/QMMû2b

2bi +QMiBMm2-

ě H2b 7QM+iBQMb `iBQMM2HH2b 2M H2b +QQ`/QMMû2b bQMi +QMiBMm2b-

ě H i`MbTQbBiBQM- H i`+2 /MbMnp(K)bQMi /2b 7QM+iBQMb +QMiBMm2b-

ě H KmHiBTHB+iBQM- H2 /ûi2`KBMMi /Mb Mn(K) bQMi /2b 7QM+iBQMb +QMiBMm2b

ě H 7QM+iBQM A 7! A ¹ = 1 det(A)

t+QK(A) 2bi +QMiBMm2 bm` GL<sub>n</sub>(K) +QKK2 7`+iBQM `iBQM@

M2HH2 2M H2b +Q2{+B2Mib /2AX 1t2KTH2b UR8V

URV , aBf ,D!R2bi +QMiBMm2 bm`D 2i TQbBiBp2 Qm MmHH2 bm` mM2 T`iB2 /2Mb2 /MbD HQ`bf 2bi TQbBiBp2 Qm MmHH2 bm`DX *QMi`2@2t2KTH2 p2+ bi`B+i2K2Mi TQbBiBp2 X

UkV , aBf , D!F 2bi +QMiBMm2 bm`D 2i MmHH2 bm` mM2 T`iB2 /2Mb2 /MbDHQ`bf 2bi MmHH2 bm`DX .2MbBiû 2i +QMiBMmBiû , T`QHQM;2K2Mi /2b BMû;HBiûb UReV

j

aQBi f +QMiBMm2 bm`Ri2HH2 [m28(x, y)2R<sup>2</sup>, f(x+y) =f(x) +f(y)X JQMi`2` [m28x2R, f(x) =f(1)·xX 1t2`+B+2 k , *HbbB[m2

aQBi f ,D!FX

URV , H 7QM+iBQMf 2bi +QMiBMm2 bm`D ,HǶBK;2 `û+BT`Q[m2 T`f /2 iQmi Qmp2`i /2F 2bi mM Qmp2`i `2HiB7 /2D

UkV , H 7QM+iBQMf 2bi +QMiBMm2 bm`D,HǶBK;2 `û+BT`Q[m2 T`f /2 iQmi 72`Kû /2F 2bi mM 72`Kû `2HiB7 /2DX

*`+iû`BbiBQM , +QMiBMmBiû ;HQ#H2 2i BK;2b `û+BT`Q[m2b /ǶQmp2`ib Qm /2 72`Kûb URdV

URV , TQm` f ,E!R+QMiBMm2- HǶ2Mb2K#H2{xi2H [m2f(x)>0}2bi Qmp2`i 2i{xi2H [m2f(x)>0}2bi 72`KûX

UkV , 1M T`iB+mHB2`-GLn(K) ={M :|det(M)|>0}2bi Qmp2`i 2i On(R) ={M :k<sup>t</sup>M M Ink60} 2bi 72`KûX úiMi #Q`Mû 2M /BK2MbBQM }MB2- BH 2bi /QM+ +QKT+iX

TTHB+iBQMb , UR3V

L2 Tb +QM7QM/`2 BK;2 `û+BT`Q[m2 2i BK;2 /B`2+i2X

PM /Bi [m2 f 2bi Qmp2`i2 bB HǶBK;2 /B`2+i2 /2 iQmi Qmp2`i /2E 2bi mM Qmp2`i /2FX G 7QM+iBQM f1:x7! bxc2bi Qmp2`i2 /2R/MbZ- KBb MǶ2bi Tb +QMiBMm2X G 7QM+iBQM f2:x7!x<sup>2</sup> 2bi +QMiBMm2 /2R/MbRKBb MǶ2bi Tb Qmp2`i2X

_2K`[m2 URNV

kV S`QHQM;2K2Mi T` +QMiBMmBiû

URV , aBf +QMp2`;2 p2`b`2F 2Ma2A\A- HQ`b HǶTTHB+iBQM f˜:A[{a}!F i2HH2 [m2f(x) =˜ f(x) bBx2A 2if(a) =˜ `2bi +QMiBMm2 2Ma2i bǶTT2HH2 T`QHQM;2K2Mi /2f T` +QMiBMmBiû 2MaX

UkV , S` mMB+Biû /2 H HBKBi2- +2 T`QHQM;2K2Mi 2bi mMB[m2X PM T`H2 /QM+/mT`QHQM;2K2Mi /2f T` +QMiBMmBiû 2Ma

.û}MBiBQM UkyV

URV , ii2MiBQM , mM2 7QM+iBQM +QMiBMm2 bm` ]a, b) MǶ/K2i Tb iQmDQm`b mM T`QHQM;2K2Mi +QMiBMm bm` [a, b)X

*@2t ,x7!sin(1/x)bm`]0,+1[X

UkV , *mHim`2 ,bBf 2bi mMB7Q`KûK2Mi +QMiBMm2 bm`A¨ pH2m`b /MbK- HQ`bf /K2i mM mMB[m2 T`QHQM;2K2Mi T` +QMiBMmBiû bm`A U2i +2 T`QHQM;2K2Mi T` +QMiBMmBiû 2bi mMB7Q`KûK2Mi +QMiBMmV

_2K`[m2b , UkRV

jV *QMiBMmBiû /2b TTHB+iBQMb HBMûB`2b 2i #BHBMûB`2b

aBf2L(E, F)- BH v û[mBpH2M+2 2Mi`2 ,

URV , f 2bi +QMiBMm2

UkV , H `2bi`B+iBQM /2f ¨B(0,1)2bi #Q`Mû2

UjV , BH 2tBbi2k2Ri2H [m2 ,8x2E-kf(x)k6kkxkX

aBf#BHBMûB`2:E1⇥E2!F- BH v û[mBpH2M+2 2Mi`2 ,

U9V , f 2bi +QMiBMm2

U8V , H `2bi`B+iBQM /2f ¨B1(0,1)⇥B2(0,1)2bi #Q`Mû2

UeV , BH 2tBbi2k2Ri2H [m2 ,

8(x, y)2E1⇥E2-kf(x, y)k6kkxkkykX h?ûQ`ĕK2 UkkV

PM MQi2 Lc(E, F) HǶ2bT+2 p2+iQ`B2H /2b TTHB+iBQMb HBMûB`2b +QMiBMm2b /2E /MbFX LQiiBQM , UkjV

URV , hQmi2 TTHB+iBQM HBMûB`2 /QMi HǶ2bT+2 /2 /ûT`i 2bi /2 /BK2MbBQM }MB2 2bi +QMiBMm2X

UkV , hQmi2 TTHB+iBQM #BHBMûB`2 /QMi H2b 2bT+2b /2 /ûT`i bQMi /2 /BK2MbBQMb }MB2b 2bi +QMiBMm2X .BK2MbBQM }MB2 2i +QMiBMmBiû Uk9V

9

.ûi2`KBM2` bB H2b TTHB+iBQMb HBMûB`2b Qm #BHBMûB`2b bmBpMi2b bQMi +QMiBMm2b ,

URV , e1: (C([0,1],R),k k1)!(R,|.|) i2HH2 [m2e1:f7!f(0)X

UkV , e1: (C([0,1],R),k k1)!(R,|.|) i2HH2 [m2e1:f7!f(0)X

UjV , p₁: (C([0,1],R),k k1)²!(C([0,1],R),k k1)i2HH2 [m2p₁: (f, g)7!f·gX

U9V , p1: (C([0,1],R),k k1)²!(C([0,1],R),k k1) i2HH2 [m2p1: (f, g)7!f·gX

U8V , :C¹([0,1],R)!C¹([0,1],R)i2HH2 [m2 (f) =f⁰ UT2m BKTQ`i2 H MQ`K2V

@ QM TQm`` `2K`[m2` [m2 iQmi2 7QM+iBQM HBMûB`2 +QMiBMm2 /QBi pQB` mM bT2+i`2 #Q`Mû @

UeV , G /û`BpiBQM /Mb(K[X],k k)p2+kPk=|P(0)|+|P⁰(0)|+|P⁰⁰(0)|+. . .

UdV , G /û`BpiBQM /Mb(K[X],k k1)p2+kPk1= sup_{[ 1,1]|P|}

U3V , JQMi`2` H +QMiBMmBiû /2b 7QM+iBQMb +QQ`/QMMû2b /Mb H #b2 +MQMB[m2 /2 K[X] 2i H /Bb+QMiBMmBiû /2b 7QM+iBQMb +QQ`/QMMû2b /Mb H #b2 (X^k/k^k)_k2NTQm` H MQ`K2 T`û+û/2Mi2X

1t2`+B+2 j , 1t2KTH2b 2i +QMi`2@2t2KTH2b 2M /BK2MbBQM BM}MB2

aQBi f2L(E, F)X JQMi`2` T` +QMi`TQbBiBQM [m2 bB TQm` iQmi2 bmBi2(un)+QMp2`;2Mi p2`b0E- H bmBi2(f(un)) 2bi #Q`Mû2- HQ`b H 7QM+iBQMf 2bi +QMiBMm2X

1t2`+B+2 9 ,

AAA ě *QMiBMmBiû bm` mM +QKT+i RV :ûMû`HBbiBQM /m i?ûQ`ĕK2 /2 HǶBK;2 +QMiBMm2 /ǶmM b2;K2Mi ,

aQBi K+QKT+i 2i f ,K !F +QMiBMm2X PM ,

URV , HǶ2Mb2K#H2 BK;2f(K)2bi +QKT+iX

UkV , bBF =R-f 2bi #Q`Mû2 2i ii2BMi b2b #Q`M2bX

1M T`iB+mHB2`- bB f 2bi +QMiBMm2 bm` H2 +QKT+iK 2i bi`B+i2K2Mi TQbBiBp2- HQ`b BH 2tBbi2 mM2 +QMbiMi2mi2HH2 [m20< m6fX

h?ûQ`ĕK2 , BK;2 +QMiBMm2 /ǶmM +QKT+i Uk8V

aQBi E=C([a, b],R)KmMB /2 H MQ`K2 /2 H +QMp2`;2M+2 mMB7Q`K2X

URV , JQMi`2` [m2 HǶ2Mb2K#H2O={f2E|8x2[a, b], f(x)>0}2bi mM Qmp2`i /2EX G2 `ûbmHii /2K2m`2@i@BH bB QM `2KTH+2[a, b]T`R\

UkV , GǶ2Mb2K#H2 /2b 7QM+iBQMb bi`B+i2K2Mi +`QBbbMi2b bm`[a, b]2bi@BH mM Qmp2`i /2E\

1t2`+B+2 8 , Pmp2`i

aQBi F= (x1, . . . , xn) mM2 7KBHH2 }MB2 /ǶûHûK2Mib /2EX GǶ2Mp2HQTT2 +QMp2t2 /2 H 7KBHH2 F 2bi T` /û}MBiBQM HǶ2Mb2K#H2 /2b #`v+2Mi`2b ¨ +Q2{+B2Mib TQbBiB7b /2x1, . . . , xn- +Ƕ2bi ¨ /B`2 H2b +QK#BMBbQMb HBMûB`2b 1x1+· · ·+

nxnp2+8i, i>02iP

i i= 1X

URV , CmbiB}2` [m2K={( 1, . . . , n)2(R+)ⁿ|P

i i= 1}2bi mM +QKT+iX

UkV , 1M /û/mB`2 [m2 HǶ2Mp2HQTT2 +QMp2t2 /ǶmM2 7KBHH2 }MB2 /2 TQBMib 2bi mM +QKT+i /2EX 1t2`+B+2 e , 1Mp2HQTT2 +QMp2t2

aQB2Mi K+QKT+i 2if:K!D+QMiBMm2 BMD2+iBp2X JQMi`2` [m2 H 7QM+iBQM `û+BT`Q[m2 2bi +QMiBMm2 bm`1t2`+B+2 d , 6QM+iBQM `û+BT`Q[m2 , f(K)X

kV *QMiBMmBiû mMB7Q`K2X V _TT2HbX

aQB2MiE2iF /2mt 2bT+2b MQ`Kûb 2if:E!F mM2 TTHB+iBQMXf 2bi /Bi2 mMB7Q`KûK2Mi +QMiBMm2 bm`E bbB , 8">0,9 >0,8(x, y)2E⇥E,(kx ykE6 ) kf(x) f(y)kF 6").

.û}MBiBQM UkeV

8

ě lM2 7QM+iBQMk@HBTb+?BixB2MM2 2bi mMB7Q`KûK2Mi +QMiBMm2X 1t2KTH2 , H 7QM+iBQM MQ`K2k kE:x7! kxkEX ě hQmi2 7QM+iBQM /û`Bp#H2 /2I⇢R!R¨ /û`Bpû2 #Q`Mû2 2bi mMB7Q`KûK2Mi +QMiBMm2 Uh6VX

ě G 7QM+iBQMx7!x² 2bi mMB7Q`KûK2Mi +QMiBMm2 bm` iQmi b2;K2Mi[a, b]- KBb Tb bm`RX ě G 7QM+iBQM `+BM2 2bi mMB7Q`KûK2Mi +QMiBMm2 bm`R+ +`8(x, y)2(R+)²,|p

x py|6p

|x y|. ě SHmb ;ûMû`H2K2Mi- QM /Bi [m2f 2bi↵@?ƺH/û`B2MM2 TQm`0<↵61bbB

9↵, C2]0,1]⇥R^⇤+,8(x, y)2E⇥E,(kf(x) f(y)kF 6Ckx yk^↵E) hQmi2 7QM+iBQM↵@?ƺH/û`B2MM2 TQm`0<↵612bi mMB7Q`KûK2Mi +QMiBMm2X

1t2KTH2b UkdV

URV , JQMi`2` [m2 HǶTTHB+iBQMf:D!F 2bi mMB7Q`KûK2Mi +QMiBMm2 bm`D bbB 8((xn)N,(yn)N)2D^N⇥D^N,

Å

yn xn !

n!+10)f(yn) f(xn) !

n!+10 ã

.

UkV , JQMi`2` [m2 H2b 7QM+iBQMb f₁:x2R7!x²- f₂ :x2R+^⇤7!1/x2if₃ :x2]0,1]7!sin(1/x)M2 bQMi Tb mMB7Q`KûK2Mi +QMiBMm2bX

UjV , JQMi`2` [m2 bBf 2bi /û`Bp#H2 bm`R+ 2i bB|f⁰|_x!+1! +1- HQ`bf MǶ2bi Tb mMB7Q`KûK2Mi +QMiBMm2X 1t2`+B+2 3 , *QKTHûK2Mi , +`+iû`BbiBQM bû[m2MiB2HH2 /2 HǶmMB7Q`K2 +QMiBMmBiû

#V :ûMû`HBbiBQM /m i?ûQ`ĕK2 /2 >2BM2 pm 2M JSaA ,

aBf:K!F 2bi mM2 7QM+iBQM +QMiBMm2 bm` mM +QKT+i K- HQ`bf 2bi mMB7Q`KûK2Mi +QMiBMm2 bm`KX h?ûQ`ĕK2 /2 >2BM2 , Uk3V

.2MbBiû /2b 7QM+iBQMb 2M 2b+HB2`b ,

aQBif ,[a, b]!F +QMiBMm2X AH 2tBbi2 mM2 bmBi2(fn)/2 7QM+iBQMb 2M 2b+HB2` i2HH2 [m2kfn fk1 !

n!10X

*2H T2`K2i 2Mi`2 mi`2 /2 /û}MB` HǶBMiû;`H2 /ǶmM2 7QM+iBQM Up2+iQ`B2HH2V +QMiBMm2 bm` mM b2;K2MiX

*Q`QHHB`2 R UkNV

+V h?ûQ`ĕK2 /ǶTT`QtBKiBQM /2 q2B2`bi`bb

.2MbBiû /2b 7QM+iBQMb TQHvMƬK2b ,

aQBif ,[a, b]!R+QMiBMm2X AH 2tBbi2 mM2 bmBi2(Pn) /2 7QM+iBQMb TQHvMQKBH2b i2HH2 [m2kPn fk1 !

n!10X

*Q`QHHB`2 k , i?ûQ`ĕK2 /2 q2B2`bi`bb UjyV

aQBif +QMiBMm2 bm`[0,1]i2HH2 [m28n2N, Z 1

0

f(x)xⁿdx= 0X JQMi`2` [m2f 2bi MmHH2X

1t2`+B+2 N , h?ûQ`ĕK2 /2b KQK2Mib

Ao ě *QMp2tBiû- +QMM2tBiû RV 1Mb2K#H2b +QMp2t2b

e

V "`v+2Mi`2b

URV , aQBi(xi)_16i6p2E^p2i( i)_16i6p2R^pi2Hb [m2Pp

i=1 i6= 0X PM TT2HH2#`v+2Mi`2/2b TQBMib TQM/û`ûb(xi, i)HǶûHûK2Mi ,

P1

i i· Xp

i=1 ixi.

UkV , 1M /BpBbMi iQmb H2b TQB/b T` mM2 +QMbiMi2 MQM MmHH2- QM M2 KQ/B}2 Tb H2 #`v+2Mi`2X .Mb iQmi2 H bmBi2- QM bmTTQb2` /QM+P

i= 1X

UjV ,bbQ+BiBpBiû , bQBi(xi, i)_16i6p2i(yj, j)_16j6q/2mt 7KBHH2b /2 TQBMib TQM/û`ûb i2HH2b [m2s1=P

i i6= 0- s2=P

jµj6= 02is1+s26= 0X

HQ`b H2 #`v+2Mi`2 /2b TQBMib TQM/û`ûb((x₁, ₁), . . .(x_p, _p),(y₁, µ₁), . . .(y_q, µ_q))2bi H2 #`v+2Mi`2 /2b #`v+2Mi`2b T`iB2Hb(b1, s1)2i(b2, s2)X

.û}MBiBQM UjRV

URV , úiMi /QMMûb /2mt p2+i2m`bx1 2ix2- QM MQi2[x1, x2]HǶ2Mb2K#H2 /2b #`v+2Mi`2b ¨ +Q2{+B2Mib TQbBiB7b /2b TQBMibx1 2ix2X AH bǶ;Bi /2b p2+i2m`bu= x1+ (1 )x2 p2+ TQbBiB75

UkV , *2ii2 /û}MBiBQM b2 ;ûMû`HBb2 /Mb H2 +b /ǶmM2 7KBHH2 /2nûHûK2MibX S` 2t2KTH2- HǶ2Mb2K#H2 /2b #`v+2Mi`2b

¨ +Q2{+B2Mib TQbBiB7b /2 i`QBb TQBMib A, B 2i C /ǶmM 2bT+2 {M2 2bi +QMbiBimû /2 HǶBMiû`B2m` ;ûQKûi`B[m2 /m i`BM;H2X

UjV , lM 2Mb2K#H2 +QMp2t2 2bi mM 2Mb2K#H2 +QMi2MMi b2b #`v+2Mi`2b ¨ +Q2{+B2Mib TQbBiB7bX

U9V , AH bm{i [mǶBH +QMiB2MM2 H2b #`v+2Mi`2b ¨ +Q2{+B2Mib TQbBiB7b /2 iQmi +QmTH2 /2 TQBMibX

"`v+2Mi`2b ¨ +Q2{+B2Mib TQbBiB7b @ 2Mb2K#H2b +QMp2t2b UjkV

URV , #QmH2- bQmb@2bT+2 {M2- /2KB@2bT+2 {M2 /Mb mMR@2p-

UkV , i`BM;H2 /Mb mM THM- TQHv;QM2b +QMp2t2b U#`v+2Mi`2b ¨ +Q2{+B2Mib TQbBiB7b /ǶmM MQK#`2 }MB /2 TQBMibV

UjV , ûTB;`T?2 /ǶmM2 7QM+iBQMf /û`Bp#H2 /2mt 7QBb bm`I i2HH2 [m2f<sup>00</sup>>0UT` 2t2KTH2 H 7QM+iBQM +``û2 Qm H 7QM+iBQM 2tTQM2MiB2HH2X

1t2KTH2b , UjjV

#V S`QT`Bûiûb 2i 2Mp2HQTT2 +QMp2t2

URV , G2b T`iB2b +QMp2t2b /2RbQMi H2b BMi2`pHH2bX

UkV , GǶBMi2`b2+iBQM /ǶmM2 7KBHH2 /2 +QMp2t2b 2bi +QMp2t2X

UjV , GǶBMi2`b2+iBQM /2 iQmi2b H2b T`iB2b +QMp2t2b +QMi2MMi mM2 T`iB2X 2bi H2 THmb T2iBi +QMp2t2 +QMi2MMiX- MQiû *QMp(X)TT2Hû2 2Mp2HQTT2 +QMp2t2 /2XX

U9V , *QMp(X)2bi HǶ2Mb2K#H2 /2b #`v+2Mi`2b ¨ +Q2{+B2Mib TQbBiB7b /2b ûHûK2Mib /2XX h?ûQ`ĕK2 Uj9V

URV , aBA2bi +QMp2t2 HQ`bA2iA˚H2 bQMi mbbBX

1t2`+B+2 Ry , AMiû`B2m` 2i /?û`2M+2 /ǶmM +QMp2t2

+V 6QM+iBQMb +QMp2t2b

aQBif:I!RX PM /Bi [m2f 2bi +QMp2t2 bm`I bB ,

8(x1, x2)2I²,8 2[0,1], f( x1+ (1 )x2)6 f(x1) + (1 )f(x2).

GǶûTB;`T?2 /ǶmM2 7QM+iBQM +QMp2t2 2bi mM +QMp2t2X

PM /Bi [m2f 2bi +QM+p2 bB 2i b2mH2K2Mi bB f 2bi +QMp2t2X .û}MBiBQM , 7QM+iBQM +QMp2t2 /2 R/MbR Uj8V

aQB2Mif 2ig /2mt 7QM+iBQMb +QMp2t2b bm`IX JQMi`2` [m2f+g 2bi mM2 7QM+iBQM +QMp2t2- KBb [m2f·g M2 HǶ2bi Tb Mû+2bbB`2K2MiX

1t2`+B+2 RR , aQKK2 2i T`Q/mBi /2 7QM+iBQMb +QMp2t2b

d

kV *QMM2tBiû T` `+b V .û}MBiBQM- 2t2KTH2b

URV , .2mt ûHûK2Mib a, b 2A ⇢ E bQMi DQB;M#H2b T` mM `+ UQm +?2KBMV /MbA bǶBH 2tBbi2 ' , [↵, ]! A

+QMiBMm2 i2HH2 [m2'(↵) =a2i'( ) =bX

UkV , AH bǶ;Bi /ǶmM2 `2HiBQM /Ƕû[mBpH2M+2 U`û~2tBp2- bvKûi`B[m2 2i i`MbBiBp2V bm` H2b ûHûK2Mib /2AX

UjV , G +QKTQbMi2 +QMM2t2 T` `+b /2a/MbA2bi HǶ2Mb2K#H2 /2b TQBMibbDQB;M#H2b ¨aT` mM `+ /MbAX

U9V , G 7KBHH2 /2b +QKTQbMi2b +QMM2t2b T` `+b /2A7Q`K2 mM2 T`iBiBQM /2AX

U8V , A2bi +QMM2t2 T` `+b bB +2ii2 7KBHH2 2bi `û/mBi2 ¨ mM2 b2mH2 +QKTQbMi2- +Ƕ2bi ¨ /B`2 [m2 /2mt ûHûK2Mib [m2H+QM[m2b /2AbQMi iQmDQm`b `2HBûb T` mM +?2KBM /MbAX

*QKTQbMi2b +QMM2t2b UjeV

URV , lM 2Mb2K#H2 +QMp2t2 2bi +QMM2t2 T` `+bX

UkV , lM 2Mb2K#H2 ûiQBHû U=`ûmMBQM /2 b2;K2Mib Bbbmb /ǶmM KāK2 TQBMiV 2bi +QMM2t2 T` `+bX (1t2KTH2 , HǶ2Mb2K#H2 /2b Ki`B+2b MBHTQi2Mi2b 2bi mM 2Mb2K#H2 ûiQBHûX)

UjV , Q2bi iQiH2K2Mi /Bb+QMiBMm U=+?[m2 +QKTQbMi2 +QMM2t2 T` `+b 2bi `û/mBi2 ¨ mM TQBMiVX

U9V , aBE 2bi mM 2bT+2 p2+iQ`B2H /2 /BK2MbBQM bmTû`B2m`2 Qm û;H2 ¨ k- HǶ2Mb2K#H2E\ {0}2bi +QMM2t2 T` `+bX

U8V , SHmb ;ûMû`H2K2Mi- bB K=C Qm bBdim(E)6= 1 2iA 2bi mM2 T`iB2 +QMp2t2 #Q`Mû2 /2E HQ`bE\A2bi +QMM2t2 T` `+bX

1t2KTH2b UjdV

#V *QMp2tBiû- +QMM2tBiû- +QMiBMmBiû

SQm`A⇢R,A2bi +QMM2t2 T` `+b()A2bi +QMp2t2()A2bi mM BMi2`pHH2X h?ûQ`ĕK2 Uj3V

URV , GǶBK;2 /ǶmM +QMM2t2 T` `+b T` mM2 7QM+iBQM +QMiBMm2 2bi +QMM2t2 T` `+bX

UkV , GǶBK;2 /ǶmM +QMM2t2 T` `+b T` mM2 7QM+iBQM +QMiBMm2 ¨ pH2m`b `û2HH2b 2bi mM BMi2`pHH2X h?ûQ`ĕK2 , BK;2 +QMiBMm2 /ǶmM +QMM2t2 T` `+b UpQB` hoAV UjNV

aQBiE mM 2bT+2 p2+iQ`B2H MQ`Kû `û2H /2 /BK2MbBQM Uûp2Mim2HH2K2Mi BM}MB2V bmTû`B2m`2 Qm û;H2 ¨ kX

URV , CmbiB}2` [m2 :x7! x

||x|| 2bi +QMiBMm2 bm`E\ {0}X

UkV , 1M /û/mB`2 [m2 H bT?ĕ`2 mMBiû 2bi +QMM2t2 T` `+bX

1t2`+B+2 Rk , aT?ĕ`2 mMBiû 2M /BK2MbBQM>2

URV , JQMi`2` [mǶBH MǶ2tBbi2 Tb /2 #BD2+iBQM +QMiBMm2 /2C/MbRX

UkV , JQMi`2` [mǶBH MǶ2tBbi2 Tb /2 #BD2+iBQM +QMiBMm2 /2U/Mb[0,1]X

1t2`+B+2 Rj , 1Mb2K#H2b MQM ?QKûQKQ`T?2b

aQB2Mig:R²!R+QMiBMm2 2iCmM +2`+H2 /2 +2Mi`2 O2i /2 `vQMR >0X

RX JQMi`2` [mǶBH 2tBbi2 /2mt TQBMibA 2iB /2C /BKûi`H2K2Mi QTTQbûb i2Hb [m2g(A) =g(B)X

kX JQMi`2` [mǶBH 2tBbi2 /2mt TQBMibC 2iD /2C- b2 /û/mBbMi HǶmM /2 HǶmi`2 T` mM [m`i /2 iQm` i2Hb [m2 g(C) =g(D)X

1t2`+B+2 R9 ,

URV , JQMi`2` [m2GLn(R)MǶ2bi Tb +QMM2t2 T` `+b /MbMn(R).

UkV , JQMi`2` [m2GLn(C)2bi +QMM2t2 T` `+b /MbMn(C).

1t2`+B+2 R8 , GLn(K)

3

o ě h`pmt /B`B;ûb

.Mb iQmi H2 T`Q#HĕK2- R /ûbB;M2 H2 +Q`Tb /2b `û2Hb- E HǶ2bT+2 p2+iQ`B2H /2b 7QM+iBQMb +QMiBMm2b- /û}MB2b bm`

[0,1]2i ¨ pH2m`b /MbR-F HǶ2bT+2 p2+iQ`B2H /2b 7QM+iBQMb /2 +Hbb2C¹- /û}MB2b bm`[0,1]2i ¨ pH2m`b /MbRX SQm` iQmif2E- QM MQi2kfk₁= sup

x2[0,1]|f(x)|2i QM `TT2HH2 [m2 HǶQM /û}MBi BMbB mM2 MQ`K2 bm`EX aBg 2bi mM2 TTHB+iBQMk 7QBb /û`Bp#H2 bm`[0,1]-g<sup>(k)</sup> /ûbB;M2 H /û`Bpû2k@BĕK2 /2gX

PM /û}MBi HǶTTHB+iBQMT [mB ¨f2E bbQ+B2 HǶTTHB+iBQMT(f)- /û}MB2 bm`[0,1]T` , 8x2[0,1], T(f)(x) =

Z x 0

f(t)dtX

SQm` iQmin2N<sup>⇤</sup>- QM MQi2T<sup>n</sup>=T T<sup>n 1</sup>- b+?Mi [m2T<sup>0</sup>=IE UTTHB+iBQM B/2MiBiû /2EVX GQ`b[m2U 2bi mM 2M/QKQ`T?BbK2 +QMiBMm /2E- QM /ûbB;M2 T`|||U|||H2 `û2H TQbBiB7 ,

|||U|||= sup

kfk₁61kU(f)k₁X Zm2biBQM yX

aQBin2N^⇤2iumM2 7QM+iBQM /2 +Hbb2Cⁿ bm`[0,1]i2HH2 [m2 ,8k2{0, . . ., n 1}, u<sup>(k)</sup>(0) = 0X RX S`Qmp2` [m2 ,8x2[0,1], u(x) =

Zx 0

(x t)^{n 1}

(n 1)! u⁽ⁿ⁾(t)dtX Zm2biBQM RX

RX JQMi`2` [m2T 2bi mM 2M/QKQ`T?BbK2 /2EX S`û+Bb2`Ker(T)2iIm(T)X Zm2H 2bi H2 bT2+i`2 /2T\ Zm2biBQM kX

RX JQMi`2` [m28x2[0,1],|T(f)(x)|6 kfk₁X

1M /û/mB`2 [m2T 2bi mM2 TTHB+iBQM +QMiBMm2 /2 (E,k k<sub>1</sub>)/Mb(E,k k<sub>1</sub>)X kX S`Qmp2` [m2|||T|||61X

jX PM +QMbB/ĕ`2f0:x2[0,1]7!f0(x) = 1X Zm2 pmikT(f0)k<sub>1</sub>\ 1M /û/mB`2|||T|||X Zm2biBQM jX

aQBin2N^⇤X

RX JQMi`2` [m2Tⁿ(f)2bi /2 +Hbb2Cⁿ bm`[0,1]X

S`Qmp2` [m2 TQm` iQmi 2MiB2` Mim`2Hk2{0, . . ., n 1}-(T<sup>n</sup>(f))<sup>(k)</sup>=T<sup>n k</sup>(f)X Zm2 pmi(T<sup>n</sup>(f))<sup>(n)</sup>\(T<sup>n</sup>(f))<sup>(k)</sup>(0)TQm` k2{0, . . ., n 1}\

kX JQMi`2` HQ`b [m2 ,8x2[0,1],(Tⁿ(f)) (x) = Z x

0

(x t)^{n 1} (n 1)! f(t)dtX jX S`û+Bb2`Ker(Tⁿ) 2iIm(Tⁿ)X

9X S`Qmp2` [m2Tⁿ 2bi mM 2M/QKQ`T?BbK2 +QMiBMm /2(E,k k<sub>1</sub>)X 8X 1M miBHBbMi H 7QM+iBQMf0 /û}MB2 ¨ H [m2biBQM kX- /ûi2`KBM2`|||Tⁿ|||.

eX S`û+Bb2` lim

n!+1TⁿX

dX S`Qmp2` [m28f2E-8x2[0,1]- H bû`B2 P

n>1(Tⁿ(f)) (x)+QMp2`;2 #bQHmK2MiX PM MQi2H HǶTTHB+iBQM [mB f2E bbQ+B2 HǶTTHB+iBQMH(f)/û}MB2 bm`[0,1]T` ,

8x2[0,1], H(f)(x) = Z x

0

e^{x t}f(t)dtX Zm2biBQM 9X

RX JQMi`2` [m2H 2bi mM 2M/QKQ`T?BbK2 /2 EX

kX S`Qmp2` [m2 ,9K>0i2H [m28x2[0,1]-|H(f)(x)|6Kkfk₁X

1M /û/mB`2 [m2H 2bi +QMiBMm2 /2(E,k k<sub>1</sub>)/Mb(E,k k<sub>1</sub>)2i [m2|||H|||6eX jX JQMi`2` [m2H(f)2biC<sup>1</sup> bm`[0,1]X *H+mH2`- TQm`x2[0,1]-(H(f))⁰(x)X

1t2`+B+2 Re , SaA 1j kyy9

N

GǶ2bT+2E =R³ 2bi KmMB /m T`Q/mBi b+HB`2 mbm2H , 8(X, Y) 2E², < X, Y >=^tXY- 2i kX k² 4 Is-s=X GǶ2bT+2E 2bi Q`B2Miû 2iB= (e1, e2, e3)2bi mM2 #b2 Q`i?QMQ`KH2 /B`2+i2 /2EX aQBiu2L(E)`2T`ûb2Miû /Mb H #b2BT` H Ki`B+2 ,

A=1 3

Ñ 2 2 1

2 1 2

1 2 2

é

RX oû`B}2` [m2A2bi mM2 Ki`B+2 Q`i?Q;QMH2 2i KQMi`2` [m2A2bi /B;QMHBb#H2 /MbM3(C)X

kX PM `TT2HH2 [m2 bBw2bi mM 2M/QKQ`T?BbK2 +QMiBMm /2 E- Uw2LC(E)V- H MQ`K2 /2 HǶ2M/QKQ`T?BbK2 w2bi /û}MB2 T` ,

k|wk|= sup

kXk=1kw(X)k

X oû`B}2` [m2u2bi mM 2M/QKQ`T?BbK2 +QMiBMm /2E 2i [m2 ,k|uk|= 1X

#X aQBi w2LC(E)X S`Qmp2` [mǶBH 2tBbi2 X02E i2H [m2kX0k= 12ik|wk|=kw(X0)kX jX PM /û}MBi /MbLC(E)H bmBi2(u^k)/2b 2M/QKQ`T?BbK2b Biû`ûb /2uX

S`Qmp2` [m2 H bmBi2(u^k) 2bi mM2 bmBi2 #Q`Mû2 /2LC(E)X 9X aQBivHǶ2M/QKQ`T?BbK2 /2E /û}MB T` ,v=id_E uX

X oû`B}2` [m2Ker(v) =V ect(1,0,1)TmBb [m2Ker(v)2iIm(v)bQMi Q`i?Q;QMmt 2iE= Ker(v) Im(v)X

#X 1M /û/mB`2 [m2-8X2E,9Z2E,9!X12Ker(v), X=X1+Z u(Z)X +X PM TQb2 TQm` iQmim2N^⇤-pm= 1

m

mX1

k=0

u^kX JQMi`2` [m2 TQm` iQmiX2E- H bmBi2(pm(X))+QMp2`;2 p2`b H T`QD2+iBQM /2X bm`Ker(v)T`HHĕH2K2Mi ¨Im(v)X

:ûMû`HBbiBQM ,

PM T`2M/ /Mb +2ii2 T`iB2E=RⁿKmMB /m T`Q/mBi b+HB`2 mbm2H , 8(X, Y)2E², < X, Y >=^tXY- H MQ`K2 /ǶmM p2+i2m` [m2H+QM[m2X /2EûiMi MQiû2kXk2i bQBiBmM2 #b2 Q`i?QMQ`KH2 /2EX

SQm` iQmi2 Ki`B+2 A = (ai,j) 2 Mn(R) [mB `2T`ûb2Mi2 /Mb H #b2 B mM 2M/QKQ`T?BbK2 u /2 E- QM MQi2 k|Ak|= sup

kXk=1kAXkX

RX oû`B}2` [m2k|k|2bi mM2 MQ`K2 bm`Mn(R)X kX Zm2H[m2b T`QT`Bûiûb /2 H Ki`B+2B=^tAAX

X JQMi`2` [m2 B2bi mM2 Ki`B+2 bvKûi`B[m2 `û2HH2X

#X PM /K2i [mǶBH 2tBbi2P 2On(R)i2HH2 [m2^tP BP bQBi /B;QMH2 `û2HH2X JQMi`2` HQ`b [m2 H2b pH2m`b T`QT`2b /2B bQMi TQbBiBp2b Qm MmHH2bX úim/B2` H2 +b QɍA2GL_n(R)X

+X PM MQi2⇢(B) = sup{| |, 2aT(B)}Qɍ aT(B)/ûbB;M2 HǶ2Mb2K#H2 /2b pH2m`b T`QT`2b /2 H Ki`B+2 BX aQBi U mM2 Ki`B+2 b2K#H#H2 ¨ BX @i@QM⇢(U) =⇢(B)\

/X JQMi`2` [m2 k|Ak|² 6⇢(B) TmBb [m2k|Ak|² =⇢(B)X

jX PM bmTTQb2 /Mb +2ii2 [m2biBQM [m2u2bi mM2 ?QKQi?ûiB2 /2 `TTQ`i 6= 0X X *H+mH2` k|Ak|X .Mb [m2H +b H bmBi2(A<sup>k</sup>)<sub>k2N</sub>2bi@2HH2 #Q`Mû2 \

#X PM bmTTQb2 [m2| |<12i QM TQb2 TQm` iQmim2N^⇤-Pm= 1 m

mX1

k=0

A^kX 1tTHB+Bi2` TQm` iQmim2N^⇤-PmTmBb /ûi2`KBM2` lim

m!+1PmX

9X PM bmTTQb2 /Mb +2ii2 [m2biBQM [m2u2bi mM 2M/QKQ`T?BbK2 bvKûi`B[m2 /2EU/QM+ /B;QMHBb#H2V 2i QM MQi2 1, . . . , nH2b pH2m`b T`QT`2b /2u- H2b iûiMi /BbiBM+ib Qm MQMX

X *H+mH2` k|Ak|X

#X .Mb [m2H +b H bmBi2(A^k)_k2N 2bi@2HH2 #Q`Mû2 \

+X PM bmTTQb2 [m2 ,8i2{1, . . . , n},| i|<12i QM TQb2 TQm` iQmim2N^⇤-Pm= 1 m

mX1

k=0

A^kX

*H+mH2` lim

m!+1PmX

1t2`+B+2 Rd , 1j JS @ kyRy

Ry

oA ě 1t2`+B+2b

GBKBi2b ,

úim/B2` H2b HBKBi2b 2M(0,0)/2b 7QM+iBQMb bmBpMi2b , RX f(x, y) = _x2^xy+y²

kX f(x, y) = ^x_{x y}^2+y2 jX f(x, y) = ^tan_x2+y^x22y

9X f(x, y) =^sin_x2^x+y^2y22

8X f(x, y) =^{1 cos(xy)}_xy2

eX f(x, y) =x^y= e^ylnx

dX f(x, y) =^sh_x+y^xshy 3X f(x, y) =^sin_shx^x _sin^shy_y ii2MiBQM ,TQm` ûim/B2`lim(x,y)!(0,0)f(x, y)- BH L1 al66Ah Sa /Ƕûim/B2`lim_x!0lim_y!0f(x, y)XXX

1t2`+B+2 R3 , GBKBi2b p2+iQ`B2HH2b

URV , _TT2H2` H2 i?ûQ`ĕK2 /m T`QHQM;2K2Mi /û`Bp#H2X

UkV , aQBif:] 1,0[[]0,+1[!R/û}MB2 T`f(x) =ln(1 +x) x x² X

JQMi`2` [m2f T2mi āi`2 T`QHQM;û2 T` +QMiBMmBiû 2M y 2i [m2 +2 T`QHQM;2K2Mi 2bi HQ`b /û`Bp#H2 2M yX Zm2HH2 2bi HQ`b H TQbBiBQM `2HiBp2 /2 H +Qm`#2 /2f T` `TTQ`i ¨ b iM;2Mi2 2M +2 TQBMi \

UjV , aQBif:R!R/û}MB2 T`f(x) =

®e ^1/x2 bBx6= 0

0 bBMQMX

JQMi`2` [m2f 2bi /2 +Hbb2C¹2i [m2 TQm` iQmin2N-f⁽ⁿ⁾(0) = 0X

*Ƕ2bi B+B mM 2t2KTH2 /2 7QM+iBQM MQM MmHH2 /QMi iQmb H2bDLn(0) bQMi MmHbX

1t2`+B+2 RN , _TT2Hb /2 JSaA

*QMiBMmBiû 2i +QMiBMmBiû mMB7Q`K2 ,

aQBi(E,k k)mM 2pM 2ia2E\ {0}X

aQBi HǶTTHB+iBQMf:E!Ri2HH2 [m2f(x) =kx akbBkxk6kak2if(x) = 0 bBkxk>kak JQMi`2` [m2f 2bi +QMiBMm2 2MaKBb [m2f MǶ2bi Tb +QMiBMm2 2M aX

1t2`+B+2 ky , 1t2KTH2 2i +QMi`2 2t2KTH2

URV , aQBi E =C([0,1],R) KmMB /2 H MQ`K2 ||.||1 /2 H +QMp2`;2M+2 2M KQv2MM2X PM TQb2'(f) = Z 1

0

f(t)dtX JQMi`2` [m2'2bi +QMiBMm2X

UkV , aQBi `<sup>1</sup> HǶ2Mb2K#H2 /2b bmBi2b `û2HH2b #Q`Mû2b KmMB /2 H MQ`K2 BM}MB2||u||1= sup_n2N|un|X aQBi /û}MB T` (u) =vQɍvn=un+1 unX JQMi`2` [m2 2bi HBMûB`2 2i +QMiBMmX

1t2`+B+2 kR , PTû`i2m`b HBMûB`2b

aQB2Mif:R!RmM2 7QM+iBQM /2 +Hbb2C¹ 2iF:R²!RH 7QM+iBQM /û}MB2 T`

F(x, y) =

®_{f(x) f(y)}

x y bBy6=x f⁰(x) bBy=x JQMi`2` [m2 H 7QM+iBQMF 2bi +QMiBMm2X

1t2`+B+2 kk ,

URV , JQMi`2` [m2 H 7QM+iBQMf:x7!ln(1 +x)2bi HBTb+?BixB2MM2 bm`R+X 1bi@2HH2 mMB7Q`KûK2Mi +QMiBMm2 \

UkV , JQMi`2` [m2 H 7QM+iBQM `+BM2 +``û2 2bi mMB7Q`KûK2Mi +QMiBMm2 bm`RKBb MǶ2bi Tb HBTb?BixB2MM2X 1t2`+B+2 kj , 6QM+iBQMb lX*X 2ifQm HBTb+?BixB2MM2

RR

*QKT+Biû

URV , JQMi`2` [m2 H bQKK2 /2 /2mt +QKT+ib 2bi mM +QKT+i

UkV , JQMi`2` [m2 HǶmMBQM /2 /2mt +QKT+ib 2bi mM +QKT+iX

QM /QMM2` /2b /ûKQMbi`iBQM /Mb H2 +b ;ûMû`H /ǶmM 2bT+2 p2+iQ`B2H /2 /BK2MbBQM [m2H+QM[m2X 1t2`+B+2 k9 , aQKK2- mMBQM ,

aQBiP 2C[X]2iF mM 72`Kû /2CX JQMi`2` [m21t2`+B+2 k8 ,P(F)2bi mM 72`Kû /2AK;2 /ǶmM 72`Kû T` mM TQHvMƬK2CX

aQBi(un)mM2 bmBi2 ¨ pH2m`b /MbE mM 2bT+2 p2+iQ`B2H /2 /BK2MbBQM }MB2X aQBiAHǶ2Mb2K#H2 /2b pH2m`b /Ƕ/?û`2M+2 /2 H bmBi2 (un)X

JQMi`2` [m2A2bi mM +QKT+i MQM pB/2 /2EX

1t2`+B+2 ke , 1Mb2K#H2 /2b pH2m`b /Ƕ/?û`2M+2

*QMM2tBiû

aQBiI mM BMi2`pHH2 /2R2if /û`Bp#H2 bm`IX PM p2mi KQMi`2` [m2f<sup>0</sup>(I) 2bi mM BMi2`pHH2X SQbQMbT={(x, y)2I², y < x}2i'(x, y) =f(y) f(x)

y x TQm` iQmi(x, y)2TX

URV , CmbiB}2` [m2'2bi #B2M /û}MB2 2i +QMiBMm2 bm`T- TmBb [m2'(T)⇢f⁰(I)⇢'(T)X

UkV , *QM+Hm`2X

1t2`+B+2 kd , h?ûQ`ĕK2 /2 .`#Qmt

URV , JQMi`2` [mǶBH MǶ2tBbi2 Tb /2 7QM+iBQM +QMiBMm2 BMD2+iBp2 /2U/MbRX

UkV , aQBiA+QMM2t2 T` `+b 2iX⇢AQmp2`i `2HiB7 2i 72`Kû `2HiB7 /2AX JQMi`2` [mǶHQ`bX=;QmX=AX 1t2`+B+2 k3 ,

JQMi`2` [m2 H2 bQmb@2Mb2K#H2 /2Mn(R)7Q`Kû /2b Ki`B+2b /B;QMHBb#H2b 2bi +QMM2t2 T` `+bX JQMi`2` [m2GLn(R)MǶ2bi Tb +QMM2t2 T` `+bX

JQMi`2` [m2GLn(C)2bi +QMM2t2 T` `+bX

JQMi`2` [m2 aP₂(R)2bi mM2 T`iB2 +QMM2t2 T` `+bX

1t2`+B+2 kN , *QMM2tBiû T` `+b

JQMi`2` [m2 H2 ;`T?2 /ǶmM2 7QM+iBQM f:IUBMi2`pHH2V!R2bi +QMM2t2 T` `+b bB 2i b2mH2K2Mi bBf2bi +QMiBMm2 bm`IX

1t2`+B+2 jy , *QMiBMmBiû 2i +QMM2tBiû T` `+b /m ;`T?2

Rk

.Bp2`b ,

aQBif: [0 ; 1]![0 ; 1]X

RX PM bmTTQb2 [m2f 2bi +QMiBMm2X JQMi`2` [m2f /K2i mM TQBMi }t2X kX PM bmTTQb2 [m2f 2bi +`QBbbMi2X JQMi`2` [m2f /K2i mM TQBMi }t2X

1t2`+B+2 jR , SQBMib }t2b

RX _TT2H2` H2 i?ûQ`ĕK2 /2 *2b¨`Q 2i H2 /ûKQMi`2`X kX aQBif:R+!RmM2 7QM+iBQM +QMiBMm2X

PM bmTTQb2 [m2lim_x!+1f(x) =`2R2i QM bQm?Bi2 ûi#HB`

x!+1lim 1 x

Zx 0

f(t) dt=`

X SQm` ">0- DmbiB}2` [mǶBH 2tBbi2A2R+ i2H [m2 TQm` iQmix>A 1

x Z x

A

(f(t) `) dt 6"

#X *QM+Hm`2 2M û+`BpMi 1 x

Z x 0

f(t) dt `=1 x

ZA 0

(f(t) `) dt+1 x

Z x A

(f(t) `) dt

1t2`+B+2 jk , *2b`Q /Bb+`2i TmBb +QMiBMm

URV , aQBi E =C([0,1],R) KmMB /2 H MQ`K2 k.k1X aQBi L mM2 7Q`K2 HBMûB`2 bm` E i2HH2 [m2 bB f >0- HQ`b L(f)>0X

JQMi`2` [m2L2bi +QMiBMm2X

UkV , JQMi`2` [mǶmM ?vT2`THM 2bi bQBi 72`Kû- bQBi /2Mb2X

UjV , JQMi`2` [mǶmM2 7Q`K2 HBMûB`2 2bi +QMiBMm2 bB 2i b2mH2K2Mi bB bQM MQvm 2bi 72`KûX

U9V , aQBi mM2 7Q`K2 HBMûB`2 MQM MmHH2 bm` E- /2 MQvm HX JQMi`2` [m2 E\H 2bi +QMM2t2 T` `+b bB 2i

b2mH2K2Mi bB MǶ2bi Tb +QMiBMm2X

1t2`+B+2 jj , 6Q`K2b HBMûB`2b

URV , aQBif2C([a, b],[a, b])X JQMi`2` [mǶBH 2tBbi2c2[a, b]i2H [m2f(c) =cX

UkV , aQBi (E,k.k) mM 2bT+2 p2+iQ`B2H MQ`Kû /2 /BK2MbBQM }MB2X aQBi K ⇢E mM +QKT+i MQM pB/2 /2 E 2i f:K!K i2HH2 [m2 ,8(x, y)2K², x6=y) kf(x) f(y)k<kx yk.

JQMi`2` [mǶBH 2tBbi2 mM mMB[m2c2K i2H [m2f(c) =cX

1t2`+B+2 j9 , SQBMi }t2 2i +QKT+i

aQBiAmM2 T`iB2 +QMp2t2 MQM pB/2 /2 R<sup>2</sup> 2if:A!RmM2 7QM+iBQM +QMiBMm2X aQBia2ib/2mt TQBMib /2A2iymM `û2H i2Hb [m2f(a)6y6f(b)X

JQMi`2` [mǶBH 2tBbi2x2Ai2H [m2f(x) =yX

1t2`+B+2 j8 , *QM@ \ @2tBiû

aQB2Mig:R²!R+QMiBMm2 2iC mM +2`+H2 /2 +2Mi`2O2i /2 `vQM R >0X

RX JQMi`2` [mǶBH 2tBbi2 /2mt TQBMibA2iB /2 C/BKûi`H2K2Mi QTTQbûb i2Hb [m2g(A) =g(B)X

kX JQMi`2` [mǶBH 2tBbi2 /2mt TQBMibC 2iD /2C- b2 /û/mBbMi HǶmM /2 HǶmi`2 T` mM [m`i /2 iQm` i2Hb [m2 g(C) =g(D)X

1t2`+B+2 je ,

Rj

úH2K2Mib /2 +Q``2+iBQM TQm` 1j kyy9 SaA , Zm2biBQM yX

RX *Ƕ2bi H 7Q`KmH2 /2 hvHQ` BMiû;`H , H T`2mp2 MǶ2bi bMb /Qmi2 Tb Mû+2bbB`2 B+B /Mb +2 bmD2i- KBb BH 2bi #QM /2 H +QMMŗi`2 , 2HH2 b2 /ûKQMi`2 T` `û+m``2M+2 bm`n2N^⇤2M miBHBbMi mM2 BMiû;`iBQM T` T`iB2bX PM `TT2HH2 m Tbb;2 [m2 HQ`b /ǶmM2 BMiû;`iBQM T` T`iB2b- BH 7mi K2MiBQMM2` UpQB`2 /ûKQMi`2`V [m2 H2b /2mt 7QM+iBQMb bQMi /2 +Hbb2C¹X SHmb T`û+BbûK2Mi- HǶBMBiBHBbiBQM /2 H `û+m``2M+2 2bi 2t+i2K2Mi HǶûMQM+û /m i?ûQ`ĕK2 7QM/K2MiH /2 HǶMHvb2- HǶ?û`û/Biû b2 7Bi 2M TQbMif(t) = (x t)ⁿ

n! -f⁰(t) = (x t)^{n 1}

(n 1)! 2ig(t) =u⁽ⁿ⁾(t), g⁰(t) =u⁽ⁿ⁺¹⁾(t)X oQB` /ûiBHb /Mb H2 +Qm`b /2 JSaAX

Zm2biBQM RX

RX PM pû`B}2 T` HBMû`Biû /2 HǶBMiû;`H2 [m2T(f+ g) =T(f) + T(g)X .2 THmb- bBf 2bi +QMiBMm2 bm`[0,1]-T(f)2bi H T`BKBiBp2 /2f [mB bǶMMmH2 2M yX .ǶT`ĕb H2 i?ûQ`ĕK2 7QM/K2MiH /2 HǶMHvb2-T(f)2bi /QM+ +QMiBMm2X .QM+T 2bi

#B2M HBMûB`2 /2E /MbEX

lM2 7QM+iBQMf2bi /MbKer(T)bbBT(f)U[mB 2bi H T`BKBiBp2 /2f[mB bǶMMmH2 2M yV 2bi MmHH2X P`f= (T(f))⁰2bi H /û`Bpû2 /2 H 7QM+iBQM MmHH2 , +Ƕ2bi /QM+ H 7QM+iBQM MmHH2X .QM+Ker(T) ={0}X

lM2 7QM+iBQMg2Im(T)bbB BH 2tBbi2f2Ei2HH2 [m2T(f) =gX .QM+g2bi mM2 T`BKBiBp2 [mB bǶMMmH2 2M yX *Ƕ2bi /QM+

/ǶT`ĕb H2 i?ûQ`ĕK2 7QM/K2MiH /2 HǶMHvb2 mM2 7QM+iBQM /2 +Hbb2C¹ [mB bǶMMmH2 2M yX

_û+BT`Q[m2K2Mi bBg 2bi mM2 7QM+iBQM /2 +Hbb2 C<sup>1</sup> [mB bǶMMmH2 2M y- iQmDQm`b /ǶT`ĕb H2 i?ûQ`ĕK2 7QM/K2MiH /2 HǶMHvb2-g=Rx

0 g⁰(t)dt=T(g⁰)X

6BMH2K2Mi-Im(T) ={/2b 7QM+iBQM /2 +Hbb2C¹ [mB bǶMMmH2Mi 2M yX}X

aQBi 2R2if2Ei2Hb [m2T(f) = f- +Ƕ2bi ¨ /B`2 2M TQbMiF =T(f),F = F<sup>0</sup>X HQ`bF(t) =C·exp( t)X JBb BH 7mi mbbBF(0) = 0- /QM+F= 02i }MH2K2Mif= 0X .QM+ MǶ2bi Tb pH2m` T`QT`2X 6BMH2K2Mi-Sp(T) =;X Zm2biBQM kX

RX 8x2[0,1], |T(f)(x)|=| Zx

0

f(t) dt 6Zx

0 |f(t)|dt6x.kfk16kfk1+`x61X .QM+8x2[0,1], |T(f)(x)|6kfk1.

1M TbbMi m bmT-kT(f)k1 6kfk1 2i T 2bi HBMûB`2 )T 2bi R@HBTb+?BixB2MM2 /QM+ +QMiBMm2 /2 (E,k.k₁) /Mb HmB@KāK2X

kX 8f2E, kT(f)k16kfk1)h

8f2E, bBkfk161, HQ`bkT(f)k161i

X .QM+ 2M TbbMi m bmT-|||T|||61 jX 8x2[0,1], T(f0)(x) =

Zx 0

dt=x) kT(f0)k1= 1X

9X kf0k1= 12ikT(f0)k1= 1)|||T|||>1- Q`|||T|||61- /ǶQɍ- T` /Qm#H2 BMû;HBiû-|||T|||= 1X Zm2biBQM jX

RX Tⁿ(f)2bi /2 +Hbb2Cⁿ, QM T2mi 7B`2 mM2 `û+m``2M+2 bm`n- 2M `2K`[mMi [m28n2N<sup>⇤</sup>, (T<sup>n</sup>(f))<sup>0</sup>=T<sup>n 1</sup>(f) (T<sup>n</sup>(f))<sup>(k)</sup> = T<sup>n k</sup>(f) , ¨k }tû- QM T2mi 7B`2 mM2 `û+m``2M+2 bm` n TQm` n > k+ 1- 2M miBHBbMi /2 MQmp2m H T`QT`Bûiû ,8p2N^⇤, (T^p(f))⁰=T^{p 1}(f)X

PM KQMi`2 mbbB [m2(T<sup>n</sup>(f))<sup>(n)</sup>=f 2i(T<sup>n</sup>(f))<sup>(k)</sup>(0) = 0 U+2 bQMi /2b T`BKBiBp2b [mB bǶMMmH2Mi 2M yXXX kX *Ƕ2bi H [m2biBQM y TTHB[mû2 ¨u=Tⁿ(f)+`(Tⁿ(f))⁽ⁿ⁾=f 2i(Tⁿ(f))^(k)(0) = 0X

jX PM pm [m2 KerT = {0} /QM+ T 2bi BMD2+iBp2 2i /QM+ 8n 2 N^⇤-Tⁿ 2bi BMD2+iBp2 +QKK2 +QKTQbû2 /2 7QM+iBQMb BMD2+iBp2bX

PM pm [m2 Im(T) 2bi HǶ2Mb2K#H2 /2b 7QM+iBQMb /2 +Hbb2 C¹ [mB bǶMMmH2 2M yX HQ`b Im(T<sup>2</sup>) 2bi HǶ2Mb2K#H2 /2b 7QM+iBQMb /QMi H /û`Bpû2 2bi /MbIm(T)2i [mB bǶMMmH2Mi 2M y , +Ƕ2bi /QM+ HǶ2Mb2K#H2 /2b 7QM+iBQMb /2 +Hbb2C²[mB bǶMMmH2Mi 2M y 2i /QMi H /û`Bpû2 bǶMMmH2 2M yX

S` `û+m``2M+2- QM KQMi`2 [m2 Im(T<sup>n</sup>) 2bi HǶ2Mb2K#H2 /2b 7QM+iBQMb /2 +Hbb2C<sup>n</sup> /QMi iQmi2b H2b /û`Bpû2b Dmb[mǶ¨

HǶQ`/`2n 1bǶMMmH2Mi 2M yX

9X Tⁿ2bi mM2 +QKTQbû2 /Ƕ2M/QKQ`T?BbK2b +QMiBMmb- /QM+ +Ƕ2bi mM 2M/QKQ`T?BbK2 +QMiBMmX 8X 1M miBHBbMi HǶû;HBiû /ǶmM2 [m2biBQM T`û+û/2Mi2 ,

8x2[0,1], |(Tⁿ(f))(x)|=| Zx

0

(x t)^{n 1}

(n 1)! f(t) dt|6(Tⁿ(|f|))(x) = Zx

0 |(x t)^{n 1} (n 1)! f(t)|dt 6Z x

0 |(x t)^{n 1}

(n 1)! |dt.kfk16

ï (x t)ⁿ n!

òx 0

.kfk16 kfk1

n! ) kTⁿ(f)k16kfk1

n! )|||Tⁿ|||6 1 n!X JBb TQm`f₀- QM , 8x2[0,1], (Tⁿ(f₀))(x) =

Z x 0

(x t)^{n 1} (n 1)! dt=

ï (x t)ⁿ n!

òx 0

= xⁿ

n! ) kTⁿ(f₀)k1= 1 n! /QM+

|||Tⁿ|||>_n!¹ 2i T` /Qm#H2 BMû;HBiû|||Tⁿ||=_n!¹X R9

eX |||Tⁿ||=_n!¹ 2ilim_n!¹ = 0X .QM+(Tⁿ)2bi mM2 bmBi2 +QMp2`;2Mi2 2ilimT<sup>n</sup>= 0<sub>L(E)</sub>X dX .ǶT`ĕb +2 [mB T`û+ĕ/2- 0 6 |Tⁿ(f)(x)| 6 kTⁿ(f)k 6 kfk

n! [mB 2bi H2 i2`K2 ;ûMû`H /ǶmM2 bû`B2 U2tTQM2MiB2HH2V +QMp2`;2Mi2X .QM+ H bû`B2P

|Tⁿ(f)(x)|2bi +QMp2`;2Mi2 /QM+ H bû`B2P

Tⁿ(f)(x)2bi #bQHmK2Mi +QMp2`;2Mi2X Zm2biBQM 9X

RX 8(f, g)2E,8(↵, )2R², H(↵f+ g) =↵H(f) + H(g)T` HBMû`Biû /2 HǶBMiû;`H2- +2 [mB MQmb /QMM2 H HBMû`Biû /2 HX

8f 2E, 8x2[0,1], H(f)(x) =e^x. Z x

0

e ^tf(t) dt- Q`x7!e<sup>x</sup> 2bi +QMiBMm2 bm`[0,1]-x7!

Z x 0

e ^tf(t) dt2bi +QMiBMm2 bm` [0,1] +QKK2 7QM+iBQM BMiû;`H2 /ǶmM2 7QM+iBQM +QMiBMm2 bm` [0,1] 2i /QM+ H(f) 2bi +QMiBMm2 +QKK2 T`Q/mBi /2 7QM+iBQMb +QMiBMm2bX /ǶQɍ8f2E, H(f)2EX

.QM+H 2bi mM 2M/QKQ`T?BbK2X kX aQBix2[0,1]X|H(f)(x)|=|

Z x 0

e^{x t}f(t) dt|6 ÅZ x 0

e ^tdt ã

| {z }

6R1

0e ^tdt=1 ¹_e

. e|{z}^x

6e

.kfk₁6(e 1).kfk₁

.QM+ TQm`K=e 1QM 8x2[0,1], |H(f)(x)|6K.kfk₁

.ǶT`ĕb HǶBMû;HBiû T`û+û/2Mi2- QM ,8f2E, kH(f)k₁6K.kfk₁ 2iH 2bi HBMûB`2- +2 [mB MQmb /QMM2 H +QMiBMmBiû /2K2M iMi [mǶTTHB+iBQM HBMûB`2 /2(E,k.k₁)/Mb HmB@KāK2 2i|||H|||6e 16eX

.QM+H 2bi +QMiBMm2 /2(E,k.k₁)/Mb HmB@KāK2 2i|||H|||6e jX SQm`f2E-8x2[0,1], H(f)(x) =e^x.Rx

0 e ^tf(t) dtX P`x7!e<sup>x</sup>2biC<sup>1</sup> bm`[0,1]- 2ix7!Rx

0 e ^tf(t) dt2biC¹ bm` [0,1]+QKK2 7QM+iBQM BMiû;`H2 /ǶmM2 7QM+iBQM +QMiBMm2 bm`[0,1]X PM 2M /û/mBi [m2H(f)2biC<sup>1</sup> bm`[0,1]2M iMi [m2 T`Q/mBi /2 7QM+iBQMb /2 +Hbb2C<sup>1</sup> bm`[0,1]X

8x2[0,1], H(f)⁰(x) =e^x. Zx

0

e ^tf(t) dt+e^x.e ^x.f(x) =H(f)(x) +f(x)X .QM+8x2[0,1], H(f)⁰(x) =H(f)(x) +f(x)X

R8