* BvmMJ - CUHK Mathematics

6mM+iBQMH MHvbBb hmiQ`BH *Hbb R

*BvmM J

*l>E

Rd a2T- kyky

*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky R f R8Tutorial 1 1 / 43

PmiHBM2

R C2Mb2MǶb BM2[mHBiv M/ uQmM;Ƕb BM2[mHBiv

k >ƺH/2`Ƕb BM2[mHBiv M/ JBMFQpbFB BM2[mHBiv

j 1tKTH2b Q7 "M+? bT+2b

9 *QmMi2`2tKTH2

*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky k f R8

:Tutorial 1 2 / 43

PmiHBM2




9 *QmMi2`2tKTH2

*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky j f R8Tutorial 1 3 / 43

C2Mb2MǶb BM2[mHBiv M/ uQmM;Ƕb BM2[mHBiv

C2Mb2MǶb BM2[mHBiv

U6BMBi2 7Q`KV 6Q` `2H +QMp2t 7mM+iBQM ϕ, MmK#2`btR, tk, · · · , tM BM Bib /QKBM- M/ TQbBiBp2 r2B;?ib B, C2Mb2MǶbBM2[mHBiv +M #2 bii2/ b

ϕ

( R∑B

∑BtB

)≤ R∑

B

∑Bϕ (tB)

UJ2bm`2@i?2Q`2iB+ M/ T`Q##BHBbiB+ 7Q`KV G2i (Ω,, µ) #2 T`Q##BHBiv bTF2 bm+? i?i µ(Ω) = R A7 ; Bb `2H pHm2/ 7mM+iBQMr?B+? Bb µ @BMi2;`#H2 M/ ϕ Bb +QMp2t 7mM+iBQM QM i?2 `2H HBM2-i?2M

ϕ

(∫

Ω;/µ

)≤

∫

Ωϕ ;/µ

*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky 9 f R8Tutorial 1 4 / 43


C2Mb2MǶb BM2[mHBivU6BMBi2 7Q`KV 6Q` `2H +QMp2t 7mM+iBQM ϕ, MmK#2`b

tR, tk, · · · , tM BM Bib /QKBM- M/ TQbBiBp2 r2B;?ib B, C2Mb2MǶbBM2[mHBiv +M #2 bii2/ b

ϕ

( R∑B

∑BtB

)≤ R∑

B

∑Bϕ (tB)


ϕ

(∫

Ω;/µ

)≤

∫

Ωϕ ;/µ

*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky 9 f R8

#

Tutorial 1 5 / 43


C2Mb2MǶb BM2[mHBivU6BMBi2 7Q`KV 6Q` `2H +QMp2t 7mM+iBQM ϕ, MmK#2`b

tR, tk, · · · , tM BM Bib /QKBM- M/ TQbBiBp2 r2B;?ib B, C2Mb2MǶbBM2[mHBiv +M #2 bii2/ b

ϕ

( R∑B

∑BtB

)≤ R∑

B

∑Bϕ (tB)


ϕ

(∫

Ω;/µ

)≤

∫

Ωϕ ;/µ


ETutorial 1 6 / 43


uQmM;Ƕb BM2[mHBiv

U7Q` T`Q/m+ibV AM biM/`/ 7Q`K- i?2 BM2[mHBiv bii2b i?i B7 , #`2 MQMM2;iBp2 `2H MmK#2`b M/ T, [ > R bm+? i?i R

T + R[ = R,

i?2M# ≤ T

T +#[

[h?2 2[mHBiv ?QH/b B7 M/ QMHv B7 T = #[XS`QQ7 , h?2 +HBK Bb +2`iBMHv i`m2 B7 = y Q` # = y. h?2`27Q`2-bbmK2 > y M/ # > y BM i?2 7QHHQrBM;X Smi i = R/T, M/ i?2M(R − i) = R/[. aBM+2 i?2 HQ;`Bi?K 7mM+iBQM Bb +QM+p2-

ln (iT + (R − i)#[) ≥ i ln (T)+(R−i) ln (#[) = ln()+ln(#) = ln(#)

rBi? i?2 2[mHBiv ?QH/BM; B7 M/ QMHv B7 T = #[X uQmM;Ƕb BM2[mHBiv7QHHQrb #v 2tTQM2MiBiBM;X_2K`F , h?2 MmK#2`b T, [ `2 bB/ iQ #2 >ƺH/2` +QMDm;i2b Q72+? Qi?2`X



uQmM;Ƕb BM2[mHBivU7Q` T`Q/m+ibV AM biM/`/ 7Q`K- i?2 BM2[mHBiv bii2b i?i B7 , #

`2 MQMM2;iBp2 `2H MmK#2`b M/ T, [ > R bm+? i?i RT + R

[ = R,i?2M

# ≤ T

T +#[

[h?2 2[mHBiv ?QH/b B7 M/ QMHv B7 T = #[X

S`QQ7 , h?2 +HBK Bb +2`iBMHv i`m2 B7 = y Q` # = y. h?2`27Q`2-bbmK2 > y M/ # > y BM i?2 7QHHQrBM;X Smi i = R/T, M/ i?2M(R − i) = R/[. aBM+2 i?2 HQ;`Bi?K 7mM+iBQM Bb +QM+p2-




← - -

⇒

Tutorial 1 8 / 43




[ = R,i?2M

# ≤ T

T +#[

[h?2 2[mHBiv ?QH/b B7 M/ QMHv B7 T = #[XS`QQ7 , h?2 +HBK Bb +2`iBMHv i`m2 B7 = y Q` # = y. h?2`27Q`2-bbmK2 > y M/ # > y BM i?2 7QHHQrBM;X

Smi i = R/T, M/ i?2M(R − i) = R/[. aBM+2 i?2 HQ;`Bi?K 7mM+iBQM Bb +QM+p2-




- .' - n i " "

Tutorial 1 9 / 43




[ = R,i?2M

# ≤ T

T +#[





E

-m m . I

Tutorial 1 10 / 43




[ = R,i?2M

# ≤ T

T +#[



rBi? i?2 2[mHBiv ?QH/BM; B7 M/ QMHv B7 T = #[X

uQmM;Ƕb BM2[mHBiv7QHHQrb #v 2tTQM2MiBiBM;X_2K`F , h?2 MmK#2`b T, [ `2 bB/ iQ #2 >ƺH/2` +QMDm;i2b Q72+? Qi?2`X


- n - n= I n a§

Tutorial 1 11 / 43




[ = R,i?2M

# ≤ T

T +#[





-

=

e r e I I-Tutorial 1 12 / 43

PmiHBM2




9 *QmMi2`2tKTH2

*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky e f R8Tutorial 1 13 / 43

>ƺH/2`Ƕb BM2[mHBiv M/ JBMFQpbFB BM2[mHBiv>ƺH/2`Ƕb BM2[mHBivU7Q` i?2 +QmMiBM; K2bm`2V A7 T, [ `2 >ƺH/2` +QMDm;i2b- i?2M

∑|tBvB| ≤

(∑|tB|T

) RT(∑

|vB|[) R

[

7Q` +QKTH2t MmK#2`b tR, tk, · · · M/ vR, vk, · · ·

S`QQ7 , qGP:- r2 bbmK2 i?i∑

|tB|T > y,∑

|vB|[ > y. G2i

=|tB|

(∑|tB|T

) RT, # =

|vB|(∑

|vB|[) R

[

TTHvBM; uQmM;Ƕb BM2[mHBiv U# ≤ T

T + #[

[ V- r2 ;2i

|tBvB|(∑

|tB|T) R

T(∑

|vB|[) R

[≤ |tB|T

T∑

|tB|T+

|vB|[

[∑

|vB|[, R ≤ B ≤ M

amKKBM; mT Qp2` B-∑

|tBvB|(∑

|tB|T) R

T(∑

|vB|[) R

[≤

∑|tB|T

T∑

|tB|T+

∑|vB|[

[∑

|vB|[=

RT +

R[ = R

U7Q` i?2 G2#2b;m2 K2bm`2V A7 Ω Bb K2bm`#H2 bm#b2i Q7 RM rBi? i?2G2#2b;m2 K2bm`2 M/ 7, ; `2 K2bm`#H2 +QKTH2t@pHm2/ 7mM+iBQMb QMΩ, i?2M

∫

Ω|7(t);(t)|/t ≤

(∫

Ω|7(t)|T/t

) RT(∫

Ω|;(t)|[/t

) R[

*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky d f R8

- f t p . a > I

#

Tutorial 1 14 / 43


∑|tBvB| ≤

(∑|tB|T

) RT(∑

|vB|[) R

[

7Q` +QKTH2t MmK#2`b tR, tk, · · · M/ vR, vk, · · ·S`QQ7 , qGP:- r2 bbmK2 i?i

∑|tB|T > y,

∑|vB|[ > y. G2i

=|tB|

(∑|tB|T

) RT, # =

|vB|(∑

|vB|[) R

[


T + #[

[ V- r2 ;2i

|tBvB|(∑

|tB|T) R

T(∑

|vB|[) R

[≤ |tB|T

T∑

|tB|T+

|vB|[

[∑

|vB|[, R ≤ B ≤ M


|tBvB|(∑

|tB|T) R

T(∑

|vB|[) R

[≤

∑|tB|T

T∑

|tB|T+

∑|vB|[

[∑

|vB|[=

RT +

R[ = R


∫

Ω|7(t);(t)|/t ≤

(∫

Ω|7(t)|T/t

) RT(∫

Ω|;(t)|[/t

) R[


O

- O F -

Tutorial 1 15 / 43


∑|tBvB| ≤

(∑|tB|T

) RT(∑

|vB|[) R

[


∑|tB|T > y,

∑|vB|[ > y. G2i

=|tB|

(∑|tB|T

) RT, # =

|vB|(∑

|vB|[) R

[


T + #[

[ V- r2 ;2i

|tBvB|(∑

|tB|T) R

T(∑

|vB|[) R

[≤ |tB|T

T∑

|tB|T+

|vB|[

[∑

|vB|[, R ≤ B ≤ M


|tBvB|(∑

|tB|T) R

T(∑

|vB|[) R

[≤

∑|tB|T

T∑

|tB|T+

∑|vB|[

[∑

|vB|[=

RT +

R[ = R


∫

Ω|7(t);(t)|/t ≤

(∫

Ω|7(t)|T/t

) RT(∫

Ω|;(t)|[/t

) R[


÷÷⇒÷i÷÷÷÷.

Tutorial 1 16 / 43


∑|tBvB| ≤

(∑|tB|T

) RT(∑

|vB|[) R

[


∑|tB|T > y,

∑|vB|[ > y. G2i

=|tB|

(∑|tB|T

) RT, # =

|vB|(∑

|vB|[) R

[


T + #[

[ V- r2 ;2i

|tBvB|(∑

|tB|T) R

T(∑

|vB|[) R

[≤ |tB|T

T∑

|tB|T+

|vB|[

[∑

|vB|[, R ≤ B ≤ M


|tBvB|(∑

|tB|T) R

T(∑

|vB|[) R

[≤

∑|tB|T

T∑

|tB|T+

∑|vB|[

[∑

|vB|[=

RT +

R[ = R


∫

Ω|7(t);(t)|/t ≤

(∫

Ω|7(t)|T/t

) RT(∫

Ω|;(t)|[/t

) R[


⇒⇒E lk 'H

4%1%4

151%1,44t . f i - =Tutorial 1 17 / 43

>ƺH/2`Ƕb BM2[mHBiv M/ JBMFQpbFB BM2[mHBivJBMFQpbFB BM2[mHBiv

U7Q` i?2 +QmMiBM; K2bm`2V 6Q` Mv T ≥ R(∑

|tB + vB|T) R

T ≤(∑

|tB|T) R

T+

(∑|vB|T

) RT

7Q` +QKTH2t MmK#2`b tR, tk, · · · M/ vR, vk, · · ·S`QQ7 UQ7 i?2 +b2 T > RV, "v >ƺH/2` BM2[mHBiv-

∑|tB + vB|T

=∑

|tB + vB| |tB + vB|T−R

≤∑

|tB| |tB + vB|T−R +∑

B|vB‖tB + vB|T−R

≤(∑

|tB|T) R

T(∑

|tB + vB|(T−R)[) R

[+(∑

|vB|T) R

T(∑

|tB + vB|(T−R)[) R

[

=(∑

|tB|T) R

T(∑

|tB + vB|T) R

[+(∑

|vB|T) R

T(∑

|tB + vB|T) R

[

.BpB/2 #Qi? bB/2b #v(∑

|tB + vB|T) R

[ M/ i?2 /2bB`2/ BM2[mHBiv 7QHHQrbX


>ƺH/2`Ƕb BM2[mHBiv M/ JBMFQpbFB BM2[mHBivJBMFQpbFB BM2[mHBiv U7Q` i?2 +QmMiBM; K2bm`2V 6Q` Mv T ≥ R

(∑|tB + vB|T

) RT ≤

(∑|tB|T

) RT+

(∑|vB|T

) RT


S`QQ7 UQ7 i?2 +b2 T > RV, "v >ƺH/2` BM2[mHBiv-∑

|tB + vB|T

=∑


≤∑



≤(∑

|tB|T) R

T(∑

|tB + vB|(T−R)[) R

[+(∑

|vB|T) R

T(∑

|tB + vB|(T−R)[) R

[

=(∑

|tB|T) R

T(∑

|tB + vB|T) R

[+(∑

|vB|T) R

T(∑

|tB + vB|T) R

[


|tB + vB|T) R



- .

. p-t,E Nitti E 51×1 t 5171 f

. p >I

Tutorial 1 19 / 43


(∑|tB + vB|T

) RT ≤

(∑|tB|T

) RT+

(∑|vB|T

) RT


∑|tB + vB|T

=∑


≤∑



≤(∑

|tB|T) R

T(∑

|tB + vB|(T−R)[) R

[+(∑

|vB|T) R

T(∑

|tB + vB|(T−R)[) R

[

=(∑

|tB|T) R

T(∑

|tB + vB|T) R

[+(∑

|vB|T) R

T(∑

|tB + vB|T) R

[


|tB + vB|T) R



I F i t¥ 7,;¥?¥÷⇒s l HittiteHilt1%1

< K Hip)Pt . (e,zµ,§(Zi = Ixitif'

E MiZ i # Z i f ) 4 ¥Tutorial 1 20 / 43


(∑|tB + vB|T

) RT ≤

(∑|tB|T

) RT+

(∑|vB|T

) RT


∑|tB + vB|T

=∑


≤∑



≤(∑

|tB|T) R

T(∑

|tB + vB|(T−R)[) R

[+(∑

|vB|T) R

T(∑

|tB + vB|(T−R)[) R

[

=(∑

|tB|T) R

T(∑

|tB + vB|T) R

[+(∑

|vB|T) R

T(∑

|tB + vB|T) R

[


|tB + vB|T) R



f t p . ta

= D - Ot.EE/XilP1Ptl5lHP#-xiti&

Tutorial 1 21 / 43


(∑|tB + vB|T

) RT ≤

(∑|tB|T

) RT+

(∑|vB|T

) RT


∑|tB + vB|T

=∑


≤∑



≤(∑

|tB|T) R

T(∑

|tB + vB|(T−R)[) R

[+(∑

|vB|T) R

T(∑

|tB + vB|(T−R)[) R

[

=(∑

|tB|T) R

T(∑

|tB + vB|T) R

[+(∑

|vB|T) R

T(∑

|tB + vB|T) R

[


|tB + vB|T) R

[ M/ i?2 /2bB`2/ BM2[mHBiv 7QHHQrbX*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky 3 f R8

"¥""

J.y.qj.gg#IaLKXil$tflyyPjp-=o-

Tutorial 1 22 / 43

>ƺH/2`Ƕb BM2[mHBiv M/ JBMFQpbFB BM2[mHBiv

JBMFQpbFB BM2[mHBivU7Q` i?2 +QmMiBM; K2bm`2V 6Q` Mv T ≥ R-

(∑|tB + vB|T

) RT ≤

(∑|tB|T

) RT+

(∑|vB|T

) RT


U7Q` i?2 G2#2b;m2 K2bm`2V A7 Ω Bb K2bm`#H2 bm#b2i Q7 RM-rBi?i?2 G2#2b;m2 K2bm`2 M/ 7, ; `2 K2bm`#H2 +QKTH2t@pHm2/7mM+iBQMb QM Ω, i?2M

(∫

Ω|7(t) + ;(t)|T/t

) RT≤

(∫

Ω|7(t)|T/t

) RT+

(∫

Ω|;(t)|T/t

) RT

*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky N f R8Tutorial 1 23 / 43



(∑|tB + vB|T

) RT ≤

(∑|tB|T

) RT+

(∑|vB|T

) RT

7Q` +QKTH2t MmK#2`b tR, tk, · · · M/ vR, vk, · · ·U7Q` i?2 G2#2b;m2 K2bm`2V A7 Ω Bb K2bm`#H2 bm#b2i Q7 RM-rBi?i?2 G2#2b;m2 K2bm`2 M/ 7, ; `2 K2bm`#H2 +QKTH2t@pHm2/7mM+iBQMb QM Ω, i?2M

(∫

Ω|7(t) + ;(t)|T/t

) RT≤

(∫

Ω|7(t)|T/t

) RT+

(∫

Ω|;(t)|T/t

) RT


PmiHBM2




9 *QmMi2`2tKTH2

*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky Ry f R8Tutorial 1 26 / 43

1tKTH2 Rs = *(E), r?2`2 E Bb +QKT+i bm#b2i BM RM. .2M2

‖7‖s = supt∈E

|7(t)|.

h?2M (s, ‖ · ‖s) Bb "M+? bT+2X

S`QQ7 , AiǶb 2bv iQ +?2+F i?i (s, ‖ · ‖s) Bb MQ`K2/ bT+2XG2iǶb T`Qp2 i?2 *QKTH2i2M2bbX amTTQb2 7M Bb *m+?v b2[m2M+2 BM s.h?2M ∀ε > y, i?2`2 2tBbib L ∈ N bm+? i?i ∀K, F > L M/ t ∈ E, Bi ?QH/bi?i

supt∈E

|7K(t)− 7F(t)| < ε.

h?2`27Q`2- 7Q` 2+? t ∈ E, 7F(t) Bb *m+?v BM RM M/ i?mb r2 +M /2M27mM+BiQM

7(t) = limF→∞

7F(t), t ∈ E.

aBM+2 L Bb BM/2T2M/2Mi Q7 t, r2 +M iF2 F → ∞ M/ +QMb2[m2MiHvsupt∈E

|7K(t)− 7(t)| < ε.

7K +QMp2`;2b mMB7Q`KHv iQ 7X >2M+2 7 ∈ *(E) #v +QKT+iM2bb Q7 E, r?B+?MBb?2b i?2 T`QQ7X

*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky RR f R8

- .

•

name,#. Hnl → I f CCK).

Tutorial 1 27 / 43


‖7‖s = supt∈E

|7(t)|.

h?2M (s, ‖ · ‖s) Bb "M+? bT+2XS`QQ7 , AiǶb 2bv iQ +?2+F i?i (s, ‖ · ‖s) Bb MQ`K2/ bT+2X

G2iǶb T`Qp2 i?2 *QKTH2i2M2bbX amTTQb2 7M Bb *m+?v b2[m2M+2 BM s.h?2M ∀ε > y, i?2`2 2tBbib L ∈ N bm+? i?i ∀K, F > L M/ t ∈ E, Bi ?QH/bi?i

supt∈E

|7K(t)− 7F(t)| < ε.


7(t) = limF→∞

7F(t), t ∈ E.


|7K(t)− 7(t)| < ε.


*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky RR f R8Tutorial 1 28 / 43


‖7‖s = supt∈E

|7(t)|.

h?2M (s, ‖ · ‖s) Bb "M+? bT+2XS`QQ7 , AiǶb 2bv iQ +?2+F i?i (s, ‖ · ‖s) Bb MQ`K2/ bT+2XG2iǶb T`Qp2 i?2 *QKTH2i2M2bbX

amTTQb2 7M Bb *m+?v b2[m2M+2 BM s.h?2M ∀ε > y, i?2`2 2tBbib L ∈ N bm+? i?i ∀K, F > L M/ t ∈ E, Bi ?QH/bi?i

supt∈E

|7K(t)− 7F(t)| < ε.


7(t) = limF→∞

7F(t), t ∈ E.


|7K(t)− 7(t)| < ε.


*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky RR f R8Tutorial 1 29 / 43


‖7‖s = supt∈E

|7(t)|.

h?2M (s, ‖ · ‖s) Bb "M+? bT+2XS`QQ7 , AiǶb 2bv iQ +?2+F i?i (s, ‖ · ‖s) Bb MQ`K2/ bT+2XG2iǶb T`Qp2 i?2 *QKTH2i2M2bbX amTTQb2 7M Bb *m+?v b2[m2M+2 BM s.h?2M ∀ε > y, i?2`2 2tBbib L ∈ N bm+? i?i ∀K, F > L M/ t ∈ E, Bi ?QH/bi?i

supt∈E

|7K(t)− 7F(t)| < ε.


- t o

Tutorial 1 30 / 43

PmiHBM2




9 *QmMi2`2tKTH2

*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky Rk f R8Tutorial 1 35 / 43

*QmMi2`2tKTH2G2i E = [y, R]. .2M2 MQ`K ‖7‖R :=

∫ Ry |7(t)|/t. h?2M

(C[y, R], ‖ · ‖R) Bb MQi "M+? bT+2X

A/2, 6BM/ *m+?v b2[m2M+2 7M- r?B+? +QMp2`;2b iQ 7mM+iBQM7 /∈ C[y, R] X*QMbB/2` i?2 b2[m2M+2

7M(t)M≥k =

y, y ≤ t ≤ Rk

M(t − R

k), R

k < t ≤ Rk + R

MR, R

k + RM < t ≤ R

AiǶb *m+?v b2[m2M+2 BM (C[y, R], ‖ · ‖R) bBM+2

‖7M − 7K‖R =Rk

∣∣∣∣RM − R

K

∣∣∣∣ → y b M,K → ∞

G2i7(t) =

y, y ≤ t ≤ R

kR, R

k < t ≤ R

*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky Rj f R8

-

F . a Cauchy sequence Hnl

tn t fo r

f a s t , but f Cf X .

Tutorial 1 36 / 43


∫ Ry |7(t)|/t. h?2M

(C[y, R], ‖ · ‖R) Bb MQi "M+? bT+2XA/2, 6BM/ *m+?v b2[m2M+2 7M- r?B+? +QMp2`;2b iQ 7mM+iBQM7 /∈ C[y, R] X

*QMbB/2` i?2 b2[m2M+2

7M(t)M≥k =

y, y ≤ t ≤ Rk

M(t − R

k), R

k < t ≤ Rk + R

MR, R

k + RM < t ≤ R


‖7M − 7K‖R =Rk

∣∣∣∣RM − R

K

∣∣∣∣ → y b M,K → ∞

G2i7(t) =

y, y ≤ t ≤ R

kR, R

k < t ≤ R


== -

Tutorial 1 37 / 43


∫ Ry |7(t)|/t. h?2M

(C[y, R], ‖ · ‖R) Bb MQi "M+? bT+2XA/2, 6BM/ *m+?v b2[m2M+2 7M- r?B+? +QMp2`;2b iQ 7mM+iBQM7 /∈ C[y, R] X*QMbB/2` i?2 b2[m2M+2

7M(t)M≥k =

y, y ≤ t ≤ Rk

M(t − R

k), R

k < t ≤ Rk + R

MR, R

k + RM < t ≤ R


-

i f I th = ,¥÷iII÷÷÷÷÷.

it:#⇒a : *Tutorial 1 38 / 43


∫ Ry |7(t)|/t. h?2M


7M(t)M≥k =

y, y ≤ t ≤ Rk

M(t − R

k), R

k < t ≤ Rk + R

MR, R

k + RM < t ≤ R


‖7M − 7K‖R =Rk

∣∣∣∣RM − R

K

∣∣∣∣ → y b M,K → ∞

G2i7(t) =

y, y ≤ t ≤ R

kR, R

k < t ≤ R


-

Tutorial 1 39 / 43


∫ Ry |7(t)|/t. h?2M


7M(t)M≥k =

y, y ≤ t ≤ Rk

M(t − R

k), R

k < t ≤ Rk + R

MR, R

k + RM < t ≤ R


‖7M − 7K‖R =Rk

∣∣∣∣RM − R

K

∣∣∣∣ → y b M,K → ∞

G2i7(t) =

y, y ≤ t ≤ R

kR, R

k < t ≤ R*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky Rj f R8

°.£"" " I¥IiI*Tutorial 1 40 / 43

*QmMi2`2tKTH2

7M(t)M≥k =

y, y ≤ t ≤ Rk

M(t − R

k), R

k < t ≤ Rk + R

MR, R

k + RM < t ≤ R


‖7M − 7K‖R =Rk

∣∣∣∣RM − R

K

∣∣∣∣ → y b M,K → ∞

G2i7(t) =

y, y ≤ t ≤ R

kR, R

k < t ≤ Rh?2M

‖7M − 7‖R =R

kM → y b M → ∞

*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky R9 f R8

-.

° ÷÷i÷÷÷

- 0 4I

Tutorial 1 41 / 43

*QmMi2`2tKTH2

7M(t)M≥k =

y, y ≤ t ≤ Rk

M(t − R

k), R

k < t ≤ Rk + R

MR, R

k + RM < t ≤ R


‖7M − 7K‖R =Rk

∣∣∣∣RM − R

K

∣∣∣∣ → y b M,K → ∞

G2i7(t) =

y, y ≤ t ≤ R

kR, R

k < t ≤ Rh?2M

‖7M − 7‖R =R

kM → y b M → ∞

>Qr2p2`- 7 /∈ C[y, R] M/ i?mb (C[y, R], ‖ · ‖R) Bb MQi "M+? bT+2X*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky R9 f R8

e -Tutorial 1 42 / 43

1M/

a22 vQm M2ti r22F

*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky R8 f R8Tutorial 1 43 / 43

* BvmMJ - CUHK Mathematics

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