* bvmmj - cuhk mathematics
TRANSCRIPT
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
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 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[mHBiv M/ uQmM;Ƕb BM2[mHBiv
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)
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 R8
#
Tutorial 1 5 / 43
C2Mb2MǶb BM2[mHBiv M/ uQmM;Ƕb BM2[mHBiv
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)
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 R8
ETutorial 1 6 / 43
C2Mb2MǶb BM2[mHBiv M/ uQmM;Ƕb BM2[mHBiv
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
*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky 8 f R8Tutorial 1 7 / 43
C2Mb2MǶb BM2[mHBiv M/ uQmM;Ƕb BM2[mHBiv
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-
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
*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky 8 f R8
← - -
⇒
Tutorial 1 8 / 43
C2Mb2MǶb BM2[mHBiv M/ uQmM;Ƕb BM2[mHBiv
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 = #[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
*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky 8 f R8
- .' - n i " "
Tutorial 1 9 / 43
C2Mb2MǶb BM2[mHBiv M/ uQmM;Ƕb BM2[mHBiv
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 = #[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
*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky 8 f R8
E
-m m . I
Tutorial 1 10 / 43
C2Mb2MǶb BM2[mHBiv M/ uQmM;Ƕb BM2[mHBiv
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 = #[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
*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky 8 f R8
- n - n= I n a§
Tutorial 1 11 / 43
C2Mb2MǶb BM2[mHBiv M/ uQmM;Ƕb BM2[mHBiv
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 = #[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
*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky 8 f R8
-
=
e r e I I-Tutorial 1 12 / 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 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
>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
O
- O F -
Tutorial 1 15 / 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
÷÷⇒÷i÷÷÷÷.
Tutorial 1 16 / 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
⇒⇒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
*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky 3 f R8Tutorial 1 18 / 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
) RT ≤
(∑|tB|T
) RT+
(∑|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
*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky 3 f R8
- .
. p-t,E Nitti E 51×1 t 5171 f
. p >I
Tutorial 1 19 / 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
) RT ≤
(∑|tB|T
) RT+
(∑|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
*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky 3 f R8
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
>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
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
*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky 3 f R8
f t p . ta
= D - Ot.EE/XilP1Ptl5lHP#-xiti&
Tutorial 1 21 / 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
) RT ≤
(∑|tB|T
) RT+
(∑|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*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
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
*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky N f R8Tutorial 1 23 / 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
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
*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky N f R8Tutorial 1 24 / 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
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
*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky N f R8Tutorial 1 25 / 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 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
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+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.
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 R8Tutorial 1 28 / 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+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.
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 R8Tutorial 1 29 / 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+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.
*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky RR f R8
- t o
Tutorial 1 30 / 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+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.
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
- - K
gt r.→ f pointwisel
y
fu. ¥ 1 convergent t o f
Tutorial 1 31 / 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+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.
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.
*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky RR f R8
@fi x ,
- I F o. -
Tutorial 1 32 / 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+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.
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
=
' ¥ ¥ ¥ I f w e c a n prove¥ 4 1Tutorial 1 33 / 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+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.
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
-
I-ftp.fmlxl-fixi/cqtmItc5tEtE-Hixi-fHeHIHmIl-tlfmlxi-fm'd
e - - ⇒ Hmm-4MTutorial 1 34 / 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 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
*QmMi2`2tKTH2G2i E = [y, R]. .2M2 MQ`K ‖7‖R :=
∫ 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
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
== -
Tutorial 1 37 / 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+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
*BvmM J U*l>EV hmiQ`BH *Hbb Rd a2T- kyky Rj f R8
-
i f I th = ,¥÷iII÷÷÷÷÷.
it:#⇒a : *Tutorial 1 38 / 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+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
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
-
Tutorial 1 39 / 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+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
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
°.£"" " 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
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 ≤ 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
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 ≤ 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