Cauchy-Schwarz 5 ”D 2ocw.nctu.edu.tw/course/vanalysis/Cauchy-Schwarz.pdf · Cauchy-Schwarz.5…”D

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Cauchy-Schwarz .��, õFK2ÌŒí

šä, éAÐ6�;Ê°çí%ð�Ó(õ7

õwf��,B'�ëËµsF �5.uÊç

bç, 7uÊ*z, *ìÜ„p�� Ó(, A

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� à‹bçí`çETGÊì2� ìÜ� „

p, ÿØ%çÞuà¤ç3� à‹¥ÿub

ç, 4�ÿ³/Bóß`í, çÞ6³/Bó

ßç (`ç�kz{… 1øZ!) /Õ}¿¡

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AøšÑ¢; Í(y�ƒDní0

ä, øF×2íöÜàBbDní

2;sz|V�˙

— Charles Harold Dodd —

Cauchy-Schwarz .��è±22D

Cauchy /É, Bb*|?cí$�Çá

1.1 ìÜ (Cauchy .��): ˛ø

26

Cauchy-Schwarz .��5…”D<2 27

a1, . . . , an, b1, . . . , bn Ñõb, †(n∑

i=1

aibi

)2

≤(

n∑i=1

a2i

)(n∑

i=1

b2i

)(1.1)

��A75k}.b‘Ku ai = λbi , i =

1, . . . , n�

¥u|?cí Cauchy .��, wõ

ç n = 3 ªJÓB¶Åbçð J. L. La-

grange (1736-1813)� Cauchy .��ªR

GBµb� àSRGá? .��É/Êõb

vn/<2, úkµbC²¾b}×üÉ[,

AÍí²ÏÿuwÅ�� úL<µb z =

x + iy, wÅ� |z| = √x2 + y2, Ä¤ú

(1.1) 7kBbÉâøIjí<2, yZÑµ

b5_b (modulus) íIj¹ª�

1.2 ìÜ (Cauchy .��): ˛ø

a1, . . . , an, b1 . . . , bn Ñµb, †∣∣∣∣∣n∑

i=1

aibi

∣∣∣∣∣2

≤(

n∑i=1

|ai|2)(

n∑i=1

|bi|2)

(1.2)

��A75k}.b‘Kuai = λbi, i =

1, . . . , n , λ Ñøµb�

I �a = (a1, a2, . . . , an)��b = (b1, b2,

. . . , bn) 1/ì2²¾5Å�,

||�a|| =√|a1|2 + · · ·+ |an|2

† Cauchy .��ª[Ñ

|�a ·�b| ≤ ‖�a‖‖�b‖ (1.3)

â(4Hb5ÜAL<£ìú˚ä³·ªJ

ì2q3, Ä¤J A = (aij) Ñø£ìú˚

ä³, †/O-5 Cauchy .��

1.3 ìÜ (Cauchy .��): ˛ø

(aij)ij Ñø£ìú˚ä³ (aij = aji),

x1, . . . , xn�y1, . . . , yn ÑL<õb (Cµb)

†∣∣∣∣∣∣n∑

i,j=1

aijxiyj

∣∣∣∣∣∣≤√√√√ n∑

i,j=1

aijxixj

√√√√ n∑i,j=1

aijyiyj

(1.4)

à‹5g)(1.4).ñqÜj, ª‚à²

¾Võ

�ξ · �η = �xA�y t =n∑

i,j=1

aijxiyj

‖ξ‖2 = �ξ · �ξ = �xA�xt =n∑

i,j=1

aijxixj

‖η‖2 = �η · �η = �yA�y t =n∑

i,j=1

aijyiyj

BbÊ¤I*7� �2, úkçbç

íA7k, ÌÌ(infinite) uø_'AÍí

–1, 1.Ûb�ŒkŠ`Cïç,¿Sí

Xx� Cauchy.��úL<AÍbn·A7,

µó~½n = ∞u´6A7á? b�J

¥_½æõÒ,�k½Ì¤Mb∑∞

i=1 |ai|2 �∑∞i=1 |bi|2 u´Ygí½æ�

1.4 ìÜ (Cauchy .��): ˛ø

ai, bi ∈ C †∣∣∣∣∣∣∞∑

i,j=1

aibj

∣∣∣∣∣∣ ≤( ∞∑

i=1

|ai|2) 1

2( ∞∑

i=1

|bi|2) 1

2

(1.5)

��A75k}.b‘Ku ai = λbi , i =

1, 2, 3 . . . , λ ∈ C� à‹∑∞

i=1 |ai|2 < ∞�∑∞i=1 |bi|2 <∞, † |∑∞

i=1 aibi| <∞�

28 bçfÈ 24»1‚ ¬89W3~

* Cauchy .��íi�7k, Ì¤b

' {ai}∞i=1 5Ij¸Yg,∑∞

i=1 |ai|2 <∞,

u'AÍ7Í|Ûí˛È, Êõ‰ƒb�C

˜ƒ}&Bb˚5Ñ l2 ˛È� ¥u n &õ

b˛È Rn |AÍíRG, …uø_ Hilbert

˛È, |½bí@àÿu¾ä‰ç�

Êbç2«wu}&çí25¬˙¦?

u

/Ì¸ ⇐⇒ Ì¤Mb ⇐⇒ 3} (1.6)

Ä¤;çÍ� Cauchy .��uªJRGB

3}�

1.5 ìÜ (Cauchy-Schwarz .��):

˛ø f, g u–È [a, b],í©/ƒb, f, g ∈C[a, b], †

∣∣∣∣∣∫ b

af(x)g(x)dx

∣∣∣∣∣2

≤∫ b

a|f(x)|2dx

∫ b

a|g(x)|2dx (1.7)

*õ‰ƒbAíi�7k, BbcÛb

° f�g uIjª3}ƒb (L2[a, b]) †

Cauchy-Schwarz .��EÍA7� w˛È

É[ªúÎ‡ø� (1.6):

Rn ⇐⇒ l2 ⇐⇒ L2 (1.8)

(1.7)¥_.��¦?˚Ñ Schwarz .

��Cauchy-Schwarz .��C Cauchy-

Schwarz-Bunyakovsky .��, uûs•

bçð Viktor Yakovlerich Bunyakovsky

(1804-1889)D,Åbçð (ŸÀš•)Karl

Herman Amandus Schwarz (1843-1921),

®Ak 1861WD 1885WêÛí� õÒ,

Bunyakovsky ª Schwarz ´bo25Wê

Û¤.��! OFí±åº??\�IÝB

\³ ! /v`F‚í±ç� ;=wõu‘

‰}0í!‹� Ê190'bçí‘‰!…,

uÊ,ÅD¶Å, ,Å†J`ê;×çD.

Š×çÑ2-�

Schwarz –ZuÊ.Š×çÿè“ç,

7(§ Kummer D Weierstrass 5�à

7^èbç� §“k Weierstrass Æ-, W

9Fk 1861Ê Weierstrass {Ð, (In-

tegral calculus) Fd5ƒ2, BDEG

O� Schwarz u Weierstrass ||Híç

Þ5ø, Fk1892WQ� Weierstrass Ê

.Š×çíP0øòƒ1917W¢� 190'í

bç!…,uµ‰ƒb�, Ä¤Ê Weier-

strass íNû-, Schwarz3buû˝\

i‰² (conformal mapping) 1−£‰

}ç (calculus of variations) Ôdu|

ü Þí½æ� FêÛàSø,šIÞøB

Öi$ít�, DÙBb˚5Ñ Schwarz-

Christoffel t�� Fyà\i‰²íxÍV

jR}j˙ (Laplace j˙) í Dirichlet

½æ, ¥j¶/^Ëfn7 Riemann ‚à

Dirichlet Ÿ† (Dirichlet principle) 5

‡� Schwarzugø_#|Æb,|¸ƒb

5 Dirichlet ½ææÊ4íÃ`„p� ¥½

æŸluâ Poisson Fj�

Schwarz|½bíiT†uk Weier-

strass 70uÞn5ýj, Fj²7#ìø|

ü Þu´ª)|üÞ3í½æ� 7Ê¥d

ı2FêÛ7ø_Ék3}í.��, ÿu

ÛÊ�2í Schwarz .��


Cauchy.��||±íRGÿu

Holder .��

1.6 ìÜ (Holder .��): ˛ø

a1, . . . , an, b1, . . . , bn ÑL<µb, /p, q ≥1 , 1

p+ 1

q= 1, †

∣∣∣∣∣n∑

i=1

aibi

∣∣∣∣∣ ≤(

n∑i=1

|ai|p) 1

p(

n∑i=1

|bi|q) 1

q

(1.9)

Holder .��ún = ∞6A7� ÇÕ|O±íÿu3}5$�

1.7 ìÜ (Holder .��): ˛øf, g ∈C[a, b], 1

p+ 1

q= 1/p, q ≥ 1†

∣∣∣∣∣∫ b

af(x)g(x)dx

∣∣∣∣∣≤(∫ b

a|f(x)|pdx

) 1p(∫ b

a|g(x)|qdx

) 1q

(1.10)

C6yøOí$�

1.8 ìÜ (Holder .��): ˛ø

f1, . . . , fn ∈ C[a, b],/ 1p1+ 1

p2+· · ·+ 1

pn=

1, pi ≥ 1 †∣∣∣∣∣∫ b

af1(x)f2(x) · · · fn(x)dx

∣∣∣∣∣≤(∫ b

a|f1(x)|p1dx

) 1p1· · ·

(∫ b

a|fn(x)|pndx

) 1pn

(1.11)

¥.��u,Åbçð Otto Lud-

wig Holder (1859-1937) Ê 1884WÈ

û˝ Fourier Mb5Yg4½æêÛí�

Holderÿèk,Å.Š×ç, §“køHŠ

�Weierstrass� KroneckerD Kummer �

A5-� Ÿlû˝E�uj&ƒb (analytic

function)D‚àmXIÌ°¸í½æ�7(

§ Kronecker D Klein í�à, 6Ipˇ

A (group theory) 5û˝, w2|O±íu

Jordan-Holder ìÜ�

Holder .��6˚Ñ Holder-Riesz

.��, ¥_.��%âª�‚bçð F.

Riesz (1880-1956) /Í$ËcÜ@à, 7

AÑû˝¡H}& (Ôdu˜ƒ}&) í3

bix� F. Riesz ªJzu˜ƒ}&5“á

6, F‚à Frechet í;¶ø Lebesgue 5

õ‰ƒbAD Hilbert-Schmidtí3}j˙

ÜAn7–©Q5=p� Ê1907-1909WÈ

F‚à Stieltjes 3})Ijª3}ƒb5

(4˜ƒí[ÛìÜ� Bb3U˚Ñ Riesz

[ÛÜA (Riesz representation theo-

rem),Fª7yû˝pŸª3}ƒb (Lp),¥

ªzuû˝ normed space(˙¸˛È) 5

Çá, w2FùªÿYg (weak conver-

gence) í–1� F6„pLp˛Èíêe4,

Dn˚Ñ Riesz-Fisher ìÜ, ¥ìÜAÑ„

p¾ä‰ç2ä³‰ç (matrix mechanics:

Heisenberg) Dš�‰ç (wave mechan-

ics: Schrodinger)s_ÜAur�íbç!

��

2. ìýì�

} Cauchy-Schwarz .��,|ßu*

ìýì�Çá, Bb5?úi$ ABC úií}¾}du

�a =−→AB, �b =

−→AC, �c =

−−→CB = �a−�b

30 bçfÈ 24»1‚ ¬89W3~

1/cq ∠CAB = θ, †âìýì� a2 +

b2 − c2 = 2ab cos θ C6²A²¾[ýÑ

|�a|2+ |�b|2−|�a−�b|2 = 2|�a||�b| cos θ (2.1)

ª)

�a ·�b = |�a||�b| cos θ (2.2)

âìýí4” | cos θ| ≤ 1, ª!A

|�a ·�b| ≤ |�a||�b| (2.3)

¥ÿu Cauchy-Schwarz .��, …µsB

bs_²¾íq3¯±ük¥s_²¾Å�

í�3�

............................................................................................................................................................................................................................

................................................................................................................................................................................................................................................... ...............

..............................................................................................................................................................................................................

A............... θ

B

C

�a

�b �c = �a−�b

Çø

¥uúk Cauchy-Schwarz .��|

òQpní«H, Ou¥³Þº/K9M)

¼ší — i�� à‹BbíE�ÉTGÊ

ù&Cú&˛È, † Cauchy-Schwarz .

��nAB¤¹—r! bçí«Ø/vu

|kJ;� úkû&CL<N &˛Èu´6

/ Cauchy-Schwarz .��á? ¤v“i

�”í<2ï,‰)æ?íkÃ, úkû&J

,5˛Èíi�uBóá? ¥_½æBb�

ƒ(2.2) �ø5ZŸÑ

cos θ =�a ·�b|�a||�b| (2.4)

˝iui�/&b (dimension) 5Ì„Í7

¬iuq3†.§¤Ì„� FJbøi�í

h1RGBò&ÝBÌ¤&˛È‘.*¬�

OG, Bb�/ø cos θ D a·b|a||b| eÑs_.

°í9Ó� ‡Þ„píj4

ìýì� =⇒ Cauchy-Schwarz .��

à‹bRG, Bb.âl[Jìýì�7ò

Q„p Cauchy-Schwarz .��:

|�a ·�b| ≤ |�a||�b|

Í(y�ƒ (2.4) Vì2i�, çÍBb}

½ÑSu cos θ 7Ýsin θ á? âk"úMü

k�k1, Ä¤s6·/ª?, OLSíRG

.ân7Êm/í9õ,Þ, FJñøí²

ÏZu cos θ�

Åj:

(1) “i�” ubç2ø'/�í3æ, Wà

Ê7Þ,àSì2á? úkyøOí

Þè6/E�ª¡©xSçjÞíùO�

úµbq3˛È, i�/s$ì2j�

cos θ =|�a ·�b|‖�a‖‖�b‖ (2.5)

cos θ =R(�a ·�b)‖�a‖‖�b‖ (2.6)

R(�a ·�b) [q3 �a ·�b 5õ¶� úk (2.5)

7kéÍ

θ =π

2⇐⇒ �a ·�b = 0

Oìýì�º.A7, O (2.6) †A7,

à‹Bborµbiíu†ªì2

cos θc =�a ·�b|�a||�b| (2.7)


(�a ·�bªJuµb),cosθc uøµbªJ[

Ñ

cos θc = ρeiϕ, ρ ≤ 1, −π ≤ ϕ ≤ π

(2.8)

w2

ρ = cos θH = | cos θc|, 0 ≤ θH ≤ π

2

θH ˚Ñ Hermitan i (Hermitian an-

gle), 7 ϕ †˚ÑÒi (pseudo-angle)�

3. Cauchy-Schwarz .��5

„p

3.1 ‡��

Ék¥_.��|?cí„pj¶u‚

à‡��, ÄÑ²¾�a��b 5Hi θ �= 0, ¹ �a

��b ÝIWís²¾, Ä¤â²¾�a��b F$A

5IÞ,L<²¾ª[Ñ

�c = �b− λ�a, λ ∈ R (3.1)

Bb5?²¾ �c íÅ�

|�c |2 = �c · �c = (�b− λ�a) · (�b− λ�a)=�b ·�b− 2�a ·�bλ+ �a · �aλ2

= |�a|2λ2 − 2�a ·�bλ+ |�b|2 (3.2)

(3.2) ªeÑ λ 5ùŸj˙�, âk |�c |2 ≥0, 7/|�a|2 ≥ 0, FJ (3.2) FH[íuÇ

¨²,7/Ê λ W,jí�Ó(, âkD λ

W.ó>, FJ³/õ;, Ä¤‡d�ükC

�k0

=(2�a·�b)2−4|�a|2|�b|2≤0=⇒|�a·�b|≤|�a||�b|

........................................................................................................................................................................................

.......................

...................................................................................

................................................... ............... |�a|2λ2−2�a ·�bλ+ |�b|2

..................................................................................................................................................................................................................................... ...............

Çù

3.2 I� — |s�×

c_„p¬˙êrfÇi�, 7Éàƒ

q3, Ä¤úkL<N&CÌ¤&˛È·ª

_à, ¥uv„pj¶|×íiõ�Í7Bb

ú¤E.Å—, Wà λ í<2uBó? Ø−

…Éuø¬¾4íix7˛ý? BbyõÇ

ø$„pj¶, EÍ"‡Þíj¶, Éu¤v

Bb¦

λ =�a ·�b|�a|2 , �c = �b− λ�a = �b− �a ·

�b

|�a|2 �a(3.3)

H� (3.2)

0≤|�c|2= |�a|2�a ·�b|�a|2

2

−2�a·�b�a ·�b|�a|2

+|�b|2(3.4)

cÜ)

(�a ·�b)2 ≤ |�a|2|�b|2

£ßu Cauchy-Schwarz .�� (Ç;U)

|�a ·�b| ≤ |�a||�b|

Ou¥_„pj¶¥7yIAÈù! ˇ

ÍÙÕìVø°,Bó·³/>H�ŸzC`

zíA.Š�L, ÿà¤ÖÈ{¬ , ççÞ

íèz.°Ýj, .;6.½|(ÿw*–

V, çbç‰Au*z, ¥õÊ.u`>íñ

32 bçfÈ 24»1‚ ¬89W3~

í, wõšàõ-2, *„p¬˙uªJä4

ø<S˜í�

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...............................�a

�c�b

λ�a

Çú

ÑS λ bà¤¦á? Bbõ²¾ �a ��c

5É[

�a · �c = �a · (�b− λ�a) = �a ·�b− λ|�a|2

Ä¤

�a · �c = 0⇐⇒ �a⊥�c⇐⇒ λ =�a ·�b|�a|2 (3.5)

²Æuz,(3.3) 2 λ í¦¶u/xS<2

1ÝÌ2Þ/ÔÑ>7Ví� ¤v λ�a £u

²¾ �b Ê �a 5I�, 7 |�c | †u �b B

�a í|s�×� λ´/ø_'|±íj„ÿ

u Lagrange�ä (Lagrange multiplier):

Ñ°�b B �a í|s�×Bb.â|cλU)

�a⊥(�b − λ�a), à‹uƒb†v‘K²Ñ}

CG� (gradient) �kÉ�

à‹Bb1.u3ObêA„p, 7I

Ñ!§„pí¬˙, CrBb}/ø<YŸ�

bçDAÞøš,¬˙ªJ}bVí½b��

ƒ (3.4)

|�a|2|�b|2 − (�a ·�b)2 = |�a|2|�c |2 (3.6)

¬i |�a|2|�c |2 ÿu Cauchy-Schwarz .�

�íÏÏá, éÍ |�a| �= 0, FJ Cauchy-

Schwarz .��bAÑ��J/ñJ

|�c | = 0 , �b = λ�a

6ÿuz�a��busIWí²¾�

û˝ç½ÿd{�.â�-�&ø¨Ø

Å,xívÈ, n}//�í9êÞ�Bb1

.3Obµs/É Cauchy–Schwarz .�

�í@à · · · ��, çbçb])àS j

�w!Z1� â (3.6) .Ø;dw˝�…”

ÿu“W��”, Ä¤BbùªGram W'�

(Gram determinant)

Γ(�a,�b)≡∣∣∣∣∣�a · �a �a ·�b�a ·�b �b ·�b

∣∣∣∣∣ , Γ(�a)≡|�a|2=�a ·�a(3.7)

Ä¤âõ �b ƒò( �aí|s�× δ ≡ |�c |ª[Ñ

δ2 =Γ(�a,�b)

Γ(�a)(3.8)

¢âk Γ(�a) = |�a|2 > 0 (ÄÑ �a �=0 ) FJï,ªJ!A Gram W'�

Γ(�a,�b) ≥ 0� çÍ‚à Gram W'�6ª

J)ƒ Cauchy-Schwarz .��í Çø$

$��

3.1 ìÜ (Cauchy-Schwarz .��):

|�a ·�b| ≤ |�a||�b| ⇐⇒ Γ(�a,�b) ≥ 0

â Gram W'�6ª‡ì²¾u´(

4Ö7�

3.2 ìÜ: ²¾ {�a,�b} (4Ö7J/ñJ Γ(�a,�b) > 0�

Åj:


(1) 3.1ìÜµsBbâä³í£ì4ªJ

û| Cauchy-Schwarz .�� âä³

ÜAVõupéí, âk

det(PP t) = (detP )2 ≥ 0 (3.9)

FJªR| Cauchy-Schwarz .��

Γ(�a,�b) =

∣∣∣∣∣(�a

�b

)(�a,�b)

∣∣∣∣∣=∣∣∣∣∣�a · �a �a ·�b�b · �a �b ·�b

∣∣∣∣∣= |�a|2|�b|2 − (�a ·�b)2 ≥ 0 (3.10)

ÝB´/yGí.��

Γ(�a,�b,�c) =

∣∣∣∣∣∣∣∣

�a

�b

�c

(�a,�b,�c)

∣∣∣∣∣∣∣∣

=

∣∣∣∣∣∣∣∣�a · �a �a ·�b �a · �c�b · �a �b ·�b �b · �c�c · �a �c ·�b �c · �c

∣∣∣∣∣∣∣∣ ≥ 0

(3.11)

Γ(�a1,�a2, . . . ,�an)

=

∣∣∣∣∣∣∣∣∣∣∣

�a1 · �a1 �a1 · �a2 . . . �a1 · �an

�a2 · �a1 �a2 · �a2 . . . �a2 · �an...

.... . .

...

�an · �a1 �an · �a2 . . . �an · �an

∣∣∣∣∣∣∣∣∣∣∣≥0

(3.12)

(2) Êƒb˛ÈWàf1, f2, . . . , fk ∈ C[a, b]† Gram W'�¦O-5$�:

Γ(f1, f2, . . . , fk) =

∣∣∣∣∣∣∣∣∣∣∣

∫ ba f

21 (x)dx

∫ ba f1(x)f2(x)dx . . .

∫ ba f1(x)fk(x)dx∫ b

a f2(x)f1(x)dx∫ ba f

22 (x)dx . . .

∫ ba f2(x)fk(x)dx

......

. . ....∫ b

a fk(x)f1(x)dx∫ ba fk(x)f2(x)dx . . .

∫ ba f

2k (x)dx

∣∣∣∣∣∣∣∣∣∣∣

†â 3.1, 3.2ìÜªø Γ(f1, f2, . . . ,

fk) ≥ 0, f1, f2, . . . , fkÑ(4óY, J

/ñJ Γ(f1, f2, . . . , fk) = 0�

(3) IÞ,L<õ(x∗, y∗)Bò(ax + by =

cí�×ªJ�âq3Vw…: íl©

Q(x∗, y∗)�(x, y)sõ/cq¤(¨Dò

(ax + by = c �ò, ÄÑ(a, b)Ñò

(ax+ by = c í¶²¾]ªcq

(x∗ − x, y∗ − y)= t(

a√a2 + b2

,b√

a2 + b2

)

,�¬j¦ÀP²¾íßTu¤vδ =

|t|u²¾ (x∗ − x, y∗ − y)5Å�Ä

Ñ(x, y)Êò(ax+ by = c,]

ax∗ + by∗ − c = t√a2 + b2

=⇒ δ= |t|= |ax∗ + by∗ − c|√a2 + b2

(3.13)

Bbøò( ax+ by = c [Ñq3:

ax+ by = (a, b) · (x, y)=√a2 + b2

c√a2 + b2

†|c|√

a2+b2£uŸõ (0, 0) Bò( ax+

by = c 5�×� °Üú&˛È,L<õ

34 bçfÈ 24»1‚ ¬89W3~

(x∗, y∗, z∗) BIÞ ax+ by+ cz = c í

�×t�

δ =|ax∗ + by∗ + cz∗ − d|√

a2 + b2 + c2(3.14)

BbøIÞ ax+ by+ cz = d [Ñq3

5$�

ax+ by + cz

= (a, b, c) · (x, y, z)=√a2 + b2 + c2

d√a2 + b2 + c2

†|d|√

a2+b2+c2uŸõ (0, 0, 0) BI

Þax+ by+ cz = d5�×�N &˛ÈE

/ó°t�

�a · �x = |�a|p (3.15)

|p|uŸõ(0, . . . , 0)B IÞ�a · �x =

|�a|p5�×� ¥$}jj¶˚ÑIÞš

(plane wave) uû˝R}j˙í½b

j¶�

.............................................................................................................................................................................................................................................................................................

....................................................................................................................................................................................................................................

................................

......................................................................................................................................................................................................................................•

ax+ by = c

δ

(x∗, y∗)

Çû

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....................

•

ax+ by + cz = d

δ

(x∗, y∗, z∗)

........

........................................................................................................................................................................................................

Çü

3.3 Þ�

(3.6)µsBbí´.ck¤, …°v6

µsBbâ²¾ �a��b F$A5IWûi$í

Þ3 A = |�a||�c| =√Γ(�a,�b), çÍâ²¾ �a

��b FHAíúi$Þ3�k

=1

2A =

1

2

√Γ(�a,�b) (3.16)

*ìýì�Võ6u'/<2í

cos θ =|�a−�b|2 − |�a|2 − |�b|2

2|�a||�b| =�a ·�b|�a||�b|

Ä¤

=1

2|�a||�b|| sin θ|

=1

2

√|�a|2|�b|2 − (�a ·�b)2

=1

2

√Γ(�a,�b) (3.17)

3.3 ìÜ: ²¾ �a��b FHíiÑ θ, †

Γ(�a,�b) = |�a|2|�b|2 sin2 θ

= |�a|2|�b|2 sin2(�a,�b) (3.18)

¥t�µsBb Gram W'�íx

S<2: Γ(�a,�b) �k²¾ �a !�b FH5


Wûi$Þ�í j� °Üú&˛Èâ²

¾ �a = (a1, a2, a3) ��b = (b1, b2, b3) �

�c = (c1, c2, c3) FH5IWýÞñ (par-

allepiped) ñ3

V ≡

∣∣∣∣∣∣∣∣a1 a2 a3

b1 b2 b3

c1 c2 c3

∣∣∣∣∣∣∣∣(3.19)

âW'�54”ª)

V 2=det

a1 a2 a3

b1 b2 b3

c1 c2 c3

a1 b1 c1

a2 b2 c2

a3 c3 c3

=

∣∣∣∣∣∣∣∣�a · �a �a ·�b �a · �c�b · �a �b ·�b �b · �c�c · �a �c ·�b �c · �c

∣∣∣∣∣∣∣∣= Γ(�a,�b,�c)

(3.20)

²ÆuzΓ(�a,�b,�c)�k²¾�a=(a1, a2, a3)�

�b = (b1, b2, b3)��c = (c1, c2, c3) FH5IW

ýÞññ35Ij�

â|s�×ªø δ = |�c | ≤ |�b|, FJâ(3.8) ªR)O±í Hadamard .��

3.4 ìÜ (Hadamard .��): Gram

W'� Γ(�a,�b)Å—.��

Γ(�a,�b) ≤ Γ(�a)Γ(�b) (3.21)

7/�UA7ík}.b‘Ku²¾ �a ��b#

ó�ò

Γ(�a,�b) = Γ(�a)Γ(�b)⇐⇒ �a⊥�b (3.22)

Hadamard.��íxS<2à-: â

²¾ �a!�b F$A5 Wûi$íÞ�ükJ

²¾ �a !�bÑsiFˇ5Åj$íÞ�, 7/

ñ% Wûi$uÅj$¹ �a⊥�b v��n

A!�

Åj:

(1) n &˛Èí Hadamard .��u

Γ(�a1,�a2, . . . ,�an)≤Γ(�a1)Γ(�a2) · · ·Γ(�an)

�ai �= 0 ∀i = 1, . . . , n (3.23)

7/�UbA7ík}.b‘Ku �a1, . . . ,

�an ssb#ó�ò, wxS<2uIWÖÞ

ñíñ3.�¬wiÅí�3�

3.4 Lagrange 0��

wõ (3.6)ÿu Lagrange 0�� (La-

grange identity):

3.5 ìÜ (Lagrange): ˛ø²¾ �a =

(a1, a2, . . . , an)� �b = (b1, b2, . . . , bn) †(n∑

i=1

a2i

)(n∑

i=1

b2i

)−(

n∑i=1

aibi

)2

=∑

1≤i<j≤n

(aibj − ajbi)2 (3.24)

C[ÑW'�

Γ(�a,�b) =

∣∣∣∣∣∣n∑

i=1a2

i

n∑i=1

aibin∑

i=1aibi

n∑i=1

b2i

∣∣∣∣∣∣=

∑1≤i<j≤n

∣∣∣∣∣ai aj

bi bj

∣∣∣∣∣2

(3.25)

„pv0��ªJ‚à¦Ñ¶‹,W'

�í4”,¥uø_'ßíæñ (à‹5/—

Dí�4!) ¥0��íxS<2u'<2�

lõ n = 2

Γ(�a,�b)=(a21+a

22)(b

21+b

22)−(a1b1+a2b2)

2

36 bçfÈ 24»1‚ ¬89W3~

=(a1b2−a2b1)2 =

∣∣∣∣∣a1 b1

a2 b2

∣∣∣∣∣2

(3.26)

ÄÑW'�

∣∣∣∣∣a1 a2

b1 b2

∣∣∣∣∣ u²¾ (a1, a2)�

(b1, b2) F$A5IWûi$íÞ3, Ä¤

Γ(�a,�b) �k (a1, a2),(b1, b2) FˇA5IW

ûi$Þ3íIj�

°ÜúL< n &Bbª!A: Γ(�a,�b)

�kâ²¾ �a = (a1, a2, . . . , an)! �b =

(b1, b2, . . . , bn) F$A5 Wûi$Þ�í

j/�kÊF%ù&è™5I�í Wû

i$Þ�í j¸� úkyøOí Gram W

'� Γ(�a1,�a2, . . . ,�an) ?/óéNít�

(~¡5ò�bç}&: M˚9O)�

Åj:

(1) ²¾Îq35Õ´/Çø_½bí¾ —

Õ3 (exterior product; vector prod-

uct)� #ì²¾ �a = (a1, a2, a3)��b =

(b1, b2, b3)ì2wÕ3

�a×�b≡

∣∣∣∣∣∣∣∣i j k

a1 a2 a3

b1 b2 b3

∣∣∣∣∣∣∣∣=

(∣∣∣ a2 a3

b2 b3

∣∣∣, ∣∣∣ a3 a1

b3 b1

∣∣∣, ∣∣∣ a1 a2

b1 b2

∣∣∣)

(3.27)

cq²¾ �a��bFHíiÑ θ, †

|�a×�b| = |�a||�b| sin θ (3.28)

FJ (3.6) C Lagrange 0�� (3.24)

µsBbq3DÕ35É[, 7w…”ÿu

H«ìÜ

sin2 θ + cos2 θ = 1

.................................................................................................................................................................................................

........................................................................................................................................................................................................................................................................ ...............

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................

................................................................................................................................................................

................................................................................................................................................................

................................................................................................................................................................

................................................................................................................................................................

(a1, a2)

(b1, b2)

A =

∣∣∣∣∣∣a1 a2

b1 b2

∣∣∣∣∣∣

Çý

.................................................................................................................................................................................................

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................................................................................................................................................................

................................................................................................................................................................

................................................................................................................................................................

................................................................................................................................................................

�a

�b

�a×�b

Çþ

4. Cauchy-Schwarz .��5

R� — Holder .��:

} Cauchy-Schwarz .��.).T

mb-xSIÌ.��, ÄÑ¥uçZ Cau-

chy íñí5ø

√ab ≤ (a+ b)⇐⇒ ab ≤ 1

2a2 +

1

2b2

(4.1)

‚à¥_.��6ªJ„p Cauchy-Sch-

warz .��: I

ai =ai√∑ni=1 a

2i

, bi =bi√∑ni=1 b

2i

H� (4.1)

aibi ≤ 1

2a2

i +1

2b2i


C

aibi√n∑

i=1a2

i

√n∑

i=1b2i

≤ 1

2

a2i√

n∑i=1

a2i

+1

2

b2i√n∑

i=1b2i

¦/Ì¸

n∑iaibi

(n∑

i=1a2

i

) 12(

n∑i=1b2i

) 12

≤ 1

2

n∑i=1a2

i

n∑i=1a2

i

+1

2

n∑i=1b2i

n∑i=1b2i

= 1

Ä¤ª)

n∑i=1

aibi ≤(

n∑i=1

a2i

) 12(

n∑i=1

b2i

) 12

ÄÑ.�� (4.1) úL<a�bîA7, à

¤orBb/”×í�4dø<‰²

ab ≤ ε

2a2 +

1

2εb2, ∀ε > 0

4.1 Young’s .��

(4.1) .��/v6˚Ñ Cauchy .�

�, wxS<2à-: 12a2, 1

2b2 }d[ýò

( y = x D x WD y WFHúi$Þ3�

7 ab †uÅj$5Þ3,*Ç$,VõéÍ

Åj$Þ3üks_úi$Þ3í¸, ÏÏ

á†uµ�í¶}� âkuÞ3, Ä¤ (4.1)

ªJ[ýÑ3}

ab ≤ 1

2a2 +

1

2b2 =

∫ a

0xdx+

∫ b

0ydy

(4.2)

7/âÇ$ñq“õ”|V��A7ík}.

b‘Ku a = b (ÄÑf = g·uÀPƒb)�

............................................................................................................................................................................................................................................................................................................ .......................

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.................................................................

a

b

12b2

12a2

Çÿ

âÇ$,}&, Bb,u/à¤íÓ;,

.�� (4.2) u´?RGBøO8$á? Ê

bç,dLSíRG5‡TuúŸl5!‹

/…”,íñ}� ú (4.2) 7k g(y) = y £

ßu f(x) = x 5¥ƒb� Ä¤ 12a2, 1

2b2 }

dus_úi$5Þ3� c_½æí�-ÿ

Êk f 5¥ƒb g u´æÊ? âƒb5Ü

ABbcÛb° f u]Óƒb (increasing

function) ¹ª�

4.1 ìÜ (Young’s .��): J y =

f(x) uø]Ó©/ƒb, Å— f(0) = 0, 7

/ x = g(y) Ñ f 5¥ƒb (Ä¤ g 6u]

Ó©/ƒb) †

ab≤∫ a

0f(x)dx+

∫ b

0g(y)dy≡Φ(a)+Φ∗(b)

(4.3)

Young’s .�� (4.3) íxS<2D‡

Þ°�Éu¤vò( y = xâ ( y = f(x)

F¦H,73}∫ b0 f(x)dx

∫ b0 g(y)dy}du

( y = f(x)(Cx = g(y))D x W y WF

H5Þ3µ�¶}EÍuÏÏá, 7/éÍ

�UA75‘Ku b = f(a) C a = g(b)�

38 bçfÈ 24»1‚ ¬89W3~

...........................................................................

b

y

a x

..........................

.......................................................

....................................................................................

...................................................................................................................................................................................................................................................................................................................................................................

y = f(x), x = g(y)

..................................................................................................................................................................................

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..

................................................................................................................................................................................................................................................................................................................................................................................... ........................................................................................................................................................................................................................................................................................................................

Ç�

|?cí Young’s .��uç f(x) =

xp−1, ÄÑ f .âu]Óƒb, Ä¤b°

p > 1, FJ g(y) = y1

p−1 ,(4.3) AÑ

ab ≤ ap

p+

b1+1

p−1

1 + 1p−1

=ap

p+bq

q(4.4)

w2 p, q Å—É[�

q = 1 +1

p− 1=

p

p− 1=

1

1− 1p

=⇒ 1

p+

1

q= 1 (4.5)

7/�UA7í‘Kub = ap−1Cap = bq�

‚à.�� (4.4) Bbï,ª) Cauchy-

Schwarz.��íRG�

4.2 ìÜ (Holder .��): #ìL<õb

a1, a2, . . . , an, b1, b2, . . . , bn /L< p, q ≥1,1

p+ 1

q= 1, †

n∑i=1

aibi ≤(

n∑i=1

|ai|p) 1

p(

n∑i=1

|bi|q) 1

q

(4.6)

„p: " (4.1) O-5„p, ¦

ai =ai

(n∑

i=1|ai|p)

1p

, bi =bi

(n∑

i=1|bi|q)

1q

,

ÄÑ Young.��úL< a, bîA7,

Ä¤ (4.4) ªJZŸAO-5$� (1p+ 1

q=

1)

ab≤ εap

p+ ε−q/p b

q

q

ab≤ 1

p(εa)p +

1

q

(b

ε

)q

ab≤ εa 1α +

(αε

) α1−α (1− α)b 1

1−α

4.2 Legendre ‰²

Bby*Çø_i�Võ Cauchy.�

� (4.1)

ab ≤ 1

2a2 +

1

2b2 ⇐⇒ a

b≤ 1

2(a

b)2 +

1

2(4.7)

Ä¤.��úL<õb a�b îA7, Ä¤

(4.7) �óçk

x ≤ 1

2x2 +

1

2, ∀x ∈ R (4.8)

........................................................

y

x

..............................................................................................................................................................................................................................................................................................................................................................

Φ(x) = 12x2

ψ(x) = (x−1) + 12

12

12

(1,0)

....................

....................

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Ç�

Bb5?ƒb

Ψ(x) =1

2x2 − x+ 1

2∀x ∈ R (4.9)


Ψ′(x) = x− 1 =⇒

Ψ′(x) > 0, x > 1

Ψ′(x) = 0, x = 1

Ψ′(x) < 0, x < 1

Ä¤ Ψ(x) 5”üM (|üM) ßÞÊ x =

1, / Ψ(1) = 0, ]

Ψ(x) =1

2x2−x+1

2≥ 0 =⇒ x ≤ 1

2x2+

1

2

yI x = abª) Cauchy .�� (4.1) 7

/�UêÞÊ Ψ(x) 5”üM x = ab= 1,

¹ a = b�

........................................................

y

x

..............................................................................................................................................................................................................................................................................................................................................................

Φ(x) = 1px

p

ψ(x) = (x−1) + 1p

1p

1q

1q

(1,0)

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...............

Ç�ø

°Üà‹ Φ(x) = 1pxp, φ(x) = x− 1

q,

p > 1; Bb5?ƒb

Ψ(x) =1

pxp − x+ 1

q∀x ∈ R (4.10)

Ψ′(x)=xp−1−1=⇒

Ψ′(x)>0, x > 1

Ψ′(x)=0, x = 1

Ψ′(x)<0, x < 1

]

Ψ(x) =1

pxp − x+ 1

q≥ 0

=⇒ x ≤ 1

pxp +

1

q

yI x = |a||b|− qp †

|a||b|− qp ≤ 1

p|a|p|b|−q +

1

q

=⇒ |a||b| ≤ 1

p|a|p + 1

q|b|q

7/�UêÞÊ

x = |a||b|− qp = 1 =⇒ |a|p = |b|q

âÇ$5}&ªøò(x = 1 Dy = x,

y = 1pxp ó>íõu (1, 1),(1, 1

p)� ò(

y = x D ( y = 1pxp ó’5(| í

�×u 1q£ßu(1, 1),(1, 1

p)sõ5�×, Ä

¤ 1p+ 1

q= 1�

*xSíi�7k,Þs_Ç$µsB

b ( y = 12x2 (Cy = 1

pxp) ¯±Êò

( y = x − 12(C y = x − 1

q) 5,j, /

y = φ(x)D y = Φ(x)ó~k (1,Φ(1)), ²

k5ò( y = φ(x) u y = Φ(x) 5X¬(

(supporting line)� à‹ø~( y = x − 12

(Cy = x − 1q) I�Ñ¦¬Ÿõ (0,0) 5ò

( y = x, † 12(C1

q) ÿuò( y = x D

( y = 12x2 (Cy = 1

pxp) ó’5(�ò�

×u 6� à‹.ÌìÊx = 1, 7ÊL<õ

x = r ‰�, †X¬(Ñ

y =Φ′(r)(x− r) + Φ(r)

= xΦ′(r)− [rΦ′(r)− Φ(r)] (4.11)

¤vyi�Ñ

rΦ′(r)− Φ(r) = maxx

(xΦ′(r)− Φ(x))

(4.12)

(4.11)µsBbâõ(Φ′(r),Φ(r)−rΦ′(r))

ªñø²ì ( y = Φ(x),íõ(r,Φ(r)),

40 bçfÈ 24»1‚ ¬89W3~

²Æuz, ¥s_õ5Èªì2/$‰²É

[, ¥ÿu O±í Legendre ‰² (Legen-

dre transformation)

Φ∗(y) = xy − Φ(x), y = Φ′(x)

(4.13)

â (4.12) ï,ª) Young’s .��

xy ≤ Φ(x) + Φ∗(y) (4.14)

Bb�ƒ‡Þs_Wä� J Φ(x) =12x2 † y = Φ′(x) = x

Φ∗(y) = xy − Φ(x) = y2 − 1

2y2 =

1

2y2

7 Young’s .��Ñ

xy ≤ Φ(x) + Φ∗(y) =1

2x2 +

1

2y2

°ÜJ Φ(x) = 1pxp, y = Φ′(x) = xp−1

Φ∗(y) = xy − Φ(x) =1

qyq

† Young’s .�� (4.14) ¦O-5$�

xy≤Φ(x)+Φ∗(y)=1

pxp+

1

qyq,

1

p+1

q=1

c_,Þ5A„FY˝íÿuX¬( (sup-

porting line) í–1, Ä¤1³/�Ì

ÊΦ(x) = 12x2� SvX¬(}æÊá ? B

b�ƒ (4.11)

(x,Φ(x))←→ (Φ′(x),Φ(x)− xΦ′(x))

5È?n7–1− 1íú@É[, FJAÍí

b°ÿuΦ′(x) uøÓƒb, à¤Φ′(x) j

ªJDõbWRn7–1 − 1 É[, ²Æu

z, Φ′′ > 0, 6ÿuΦ uøoƒb (convex

function) † Young’s .��A7� ¥T~

Bb Holder .��ªJ*oƒbíi�V

25, 7w…”ÿu Jensen .��

BbªJ…<[ýà- (‚àoƒb5

4”)

ab= elog ab = elog a+log b = e12

log a2+ 12

log b2

≤ 1

2e

12

log a2

+ e12

log b2 =1

2a2 +

1

2b2

(4.15)

°Ü

ab= elog ab = elog a+log b = e1p

log ap+ 1q

log bq

≤ 1

pelog ap

+ elog bq

=1

pap +

1

qbq (4.16)

¥s_.��íxS<2ªJâNbƒbex

5Ç$VÜj,Ñ7jZ–cBbcqa < b,

log a < log b âkex uøoƒb, FJ©

Q (log a2, a2)� (log b2, b2) C (log ap,

ap)�(log bq, bq) sõíýøìrÊ©Q¥s

õíC (ƒbex Ç$5ø¶M) 5,j� Ä

ÑBbı=.��

log ap ≤ log ab ≤ log bq

A7, Ä¤p� q .âÅ—p > 1� q > 1� â

}õt� log ab = 1plog ap + 1

qlog bqªJ

[Ñ log ap� log bq ío ¯ (convex com-

bination), Ä¤p� q´bÅ— 1p+ 1

q= 1, ¢

âG$ (¨Öý5¶M), ‚àú@iAªW,

FJyâ}õt�ªøÊ ab£,jýµõí

MÑ1pap + 1

qbq, Ä¤ªJ!A

ab ≤ 1

pap +

1

qbq


...........................................................................................................................................................................................................................................................................................................................................................................................................................................

ab

12 (a2 + b2)

a2b2

log a2 log b2log ab

= 12 log a2 + 1

2 log b2

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Ç�ù

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ab

1pa

p + 1q b

q

apbq

log ap log bqlog ab

= 1p log ap + 1

q log bq

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Ç�ú

|(Bb*¾å}& ( dimensional

analysis) íi�VzpÑSp� q.âÅ—1p+ 1

q= 1� ílø x ‰²Ñ λx, 1ì2

ƒb fλ(x) ≡ f(λx) †

‖fλ‖≡(∫|fλ(x)|pdx

) 1p

=(1

λ

) 1p(∫

f(λx)dλx) 1

p

=1

λ1p

‖f‖p

Bb…<cq‖fλ‖p = 1, †

‖f‖p ←→ λ1p ←→ 1

p

°Ü

‖g‖q ←→ λ1q ←→ 1

q

ÄÑ

∫|fλ(x)gλ(x)|dx

=1

λ

∫f(λx)g(λx)dλx

=1

λ

∫f(y)g(y)dy

Ä¤ Holder .�� (1.10) bA7íuw˝

¬s�5¾å (dimension) .âó°

1

λ=(1

λ

) 1p(1

λ

) 1q

⇐⇒ 1

p+

1

q= 1

²Æuzf5¾åªeÑ1p, 7g 5¾å†

Ñ1q, 3}

∫ |fg|dx b/<2, 6ÿuz3}

uøõb, 7õbuø&]1p+ 1

q= 1� °Ü

(1.11) ˝¬s�5¾å.âó°]

1

p1+

1

p2+ · · ·+ 1

pn= 1

Holder .��Çø_RG5��Ñ

4.3 ìÜ: ˛ø f, g ∈ C[a, b] / 1 ≤p, q, r <∞, 1

p+ 1

q= 1

r, †

( ∫ b

a|f(x)g(x)|rdx

) 1r

≤( ∫ b

a|f(x)|pdx

) 1p( ∫ b

a|g(x)|qdx

) 1q

C

‖fg‖r ≤ ‖f‖p‖g‖q¥_„pu'ßí3æ, BbG#è6

d*3� â¾å}&ª)

‖fλgλ‖r =(1

λ

) 1r ‖fg‖r

42 bçfÈ 24»1‚ ¬89W3~

FJ.��˝ií¾å (dimension) �k1r,

7¬iu 1p+ 1

q, FJ p, q, r.âÅ—É[

�1p+ 1

q= 1

r�

|(Bbb#|íu Holder .��5

…”ÿuoƒb

4.4 ìÜ: ì2ƒb

F (f) ≡ ln( ∫ b

aef(x)dx

)

†Fuøoƒb�

„p: â Holder .��¦ p = 1θ†

F (θf + (1− θ)g)= ln

[ ∫ b

aeθf(x)e(1−θ)g(x)dx

]

≤ ln[( ∫ b

aef(x)dx

)θ( ∫ b

aeg(x)dx

)1−θ]

= θF (f) + (1− θ)F (g)

Åj:

(1) * Holder .��|êª)ƒrÖ½

b//�í.��, w2||±íÿu

˛ÈDmäí�M½æ (interpolation

problem)� Wà Riesz convex ìÜ,

Riesz-Torin ìÜC Riesz-Torin-Stein

ìÜ · · · ��

¡5’e:

1. A. Cauchy, Cours d’Analyse de EcoleRoyal Polytechnique, 1821.

2. G. H. Hardy, J. E. Littlewood and G.Polya, Inequalities, Cambridge Univer-sity Press, Cambridge, 1952.

3. O. Holder, Uber einen Mittelwertsatz,Gottingen Nachr., 1889.

4. F. Riesz, Untersuchungen uber Sys-teme integrierbarer Funktionen, Math.Ann., 69 (1910).

5. H. A. Schwarz, Zur Integration der par-tiellen Differentialgleichung ∂2u/∂x2 +∂2u/∂y2 = 0, Coll. Works, Berlin,1890.

6. T. Needham, A visual explanation ofJensen’s inequality, American Math.Monthly 100 (1993), 768-771.

7. M˚9O, ò�bç}&, {æ|#þ,1987�

8. ŠœþO, oƒb, Jensen .��D Leg-endre ‰², bçfÈ (2Ûû˝ÍbçF),Vol. 76, p.51-57, 1995�

—…dT6ÛL`kÅ!AŠ×çbçÍ�

@àbçû˝F—

Documents

Cauchy-Schwarz 5 ”D 2ocw.nctu.edu.tw/course/vanalysis/Cauchy-Schwarz.pdf · Cauchy-Schwarz.5…”D