huongdantuhoc-giaitichham-09-09-09

8/14/2019 HuongDanTuHoc-GiaiTichHam-09-09-09

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Hng dn t hc Gii tch hmLp Ton 2006 AB

Nguyn Vn DngKhoa Ton hc

i hc ng ThpEmail: [email protected]

Ngy 09 thng 09 nm 2009


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Tm tt ni dung

Ti liu ny c bin son cho sinh vin t hc v t nghin cu mn hc Gii tch hmtheo cun sch Gii tch hm ca tc gi u Th Cp, NXB Gio dc 2008, Th vin KhoaTon hc v Th vin Trng i hc ng Thp.


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Mc lc

cng mn hc 3

0 Kin thc chun b 70.1 i s tuyn tnh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Hng dn t hc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70.2 Khng gian mtric v khng gian tp . . . . . . . . . . . . . . . . . . . . . 7

Hng dn t hc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70.3 Bi tp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1 Khng gian nh chun v khng gian Banach 81.1 Khng gian nh chun v khng gian Banach . . . . . . . . . . . . . . . . . 8

Hng dn t hc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2 Chui trong khng gian nh chun . . . . . . . . . . . . . . . . . . . . . . . 8

Hng dn t hc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Khng gian Lp(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Hng dn t hc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 nh x tuyn tnh lin tc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Hng dn t hc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Khng gian con v khng gian thng . . . . . . . . . . . . . . . . . . . . . 10

Hng dn t hc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6 Khng gian chiu v khng gian kh li . . . . . . . . . . . . . . . . . . . . . 11

Hng dn t hc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Cc nh l c bn ca gii tch hm 122.1 nh l Hahn-Banach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Hng dn t hc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 nh l nh x m v nh l th ng . . . . . . . . . . . . . . . . . . . . 12

Hng dn t hc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Nguyn l b chn u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Hng dn t hc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Khng gian cc hm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Hng dn t hc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Khng gian Hilbert 143.1 Tch v hng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Hng dn t hc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 H trc giao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Hng dn t hc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 Tng Hilbert ca cc khng gian Hilbert . . . . . . . . . . . . . . . . . . . . 15Hng dn t hc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

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3.4 H trc chun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Hng dn t hc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 L thuyt ton t 174.1 Khng gian lin hp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Hng dn t hc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2 Ton t compc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Hng dn t hc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.3 Ton t b chn trong khng gian Hilbert . . . . . . . . . . . . . . . . . . . . 17

Hng dn t hc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17


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cngKHOA TON HC CNG HO X HI CH NGHA VIT NAM

B MN GII TCH c lp T do Hnh phc

CNG CHI TIT HC PHN NM HC 2009 - 2010

1. Tn hc phn

Gii tch hm

2. S n v hc trnh6 n v hc trnh = 90 tit

3. Trnh i hc S phm Ton hc

Lp Ton 2006AB

4. Phn b thi gian- L thuyt: 60 tit- Thc hnh (bi tp, kim tra, tho lun, xmina, ...): 30 tit

5. iu kin tin quyt o-Tch phn

6. Mc tiu ca hc phn- Trang b cho sinh vin nhng kin thc c bn v khng gian nh chun, khng gianBanach, mt s nh l c bn ca gii tch hm, khng gian Hilbert, khng gian cchm lin tc v k nng c bn trong vic gii quyt cc bi tp lin quan- Lm r c mi lin h gia cc hc phn gii tch ton hc, qua gip cho sinhvin hiu c s pht trin v vai tr ca cc hc phn gii tch ton

7. M t vn tt ni dung ca hc phn- Khng gian nh chun- Mt s nh l c bn ca gii tch hm

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- Khng gian Hilbert- Khng gian cc hm lin tc

8. Nhim v ca sinh vin

- Tham d y cc bui ln lp ca ging vin- Lm bi tp v tham d y cc k thi, kim tra, cc bui tho lun, ...

9. Ti liu hc tp- Ti liu tham kho chnh[1] u Th Cp, Gii tch hm, NXB Gio dc, 2002- Ti liu tham kho khc[1] Nguyn Xun Lim, Gii tch hm, NXB Gio dc, 2001.

[2] Nguyn Vn Khu, L Mu Hi, C s l thuyt hm v Gii tch hm, Tp 2, NXBGio dc, 2001.

10. Tiu chun nh gi sinh vin- Kim tra thng xuyn: 2 bi- Thi kt thc hc phn: 1 bi (120 pht)

11. Thang im

- Kim tra thng xuyn: 30%- Thi kt thc hc phn: 70%

12. Ni dung chi tit hc phn

Chng 1 KHNG GIAN NH CHUN V KHNG GIAN BANACH(20 tit l thuyt + 10 tit thc hnh)

1. Cc khi nim c bn- nh ngha chun v khng gian nh chun- Cc v d- Tp li v tp b chn

2. Chui trong khng gian nh chun- Chui v s hi t- Chui hi t tuyt i

3. Khng gian Lp(X) v lp- nh ngha- Tnh cht

4. nh x tuyn tnh lin tc

- nh x tuyn tnh lin tc- Khng gian L(E, F)

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5. Khng gian con v khng gian thng- Khng gian con- Khng gian thng

6. Khng gian hu hn chiu, khng gian kh li- Khng gian hu hn chiu- Khng gian kh li

Chng 2 CC NH L C BN CA GII TCH HM(10 tit l thuyt + 10 tit thc hnh)

1. nh l Hahn-Banach- S chun, na chun v b Zorn- nh l Hahn-Banach- H qu ca nh l Hahn-Banach

2. nh l nh x m- nh l Baire v phm trnh l nh x m- H qu ca nh l nh x m

3. nh l th ng- th ca nh x- th ng

4. Nguyn l b chn u- Na chun lin tc- Nguyn l b chn u

Chng 3 KHNG GIAN HILBERT(15 tit l thuyt + 10 tit thc hnh)

1. Cc khi nim c bn- Tch v hng v khng gian Hilbert- ng thc bnh hnh

2. H trc giao- H trc giao- Php chiu trc giao- Phim hm tuyn tnh trn khng gian Hilbert

3. Tng Hilbert ca cc khng gian Hilbert- Tng Hilbert ca cc khng gian Hilbert- Tng Hilbert ca cc khng gian con ng

4. H trc chun- H trc chun- Khai trin trc chun

Chng 4 KHNG GIAN CC HM LIN TC(10 tit l thuyt + 5 tit thc hnh)

1. Khng gian cc hm- Khng gian cc hm b chn- Khng gian cc hm lin tc- Cc loi hi t trong khng gian hm

2. nh l Stone - Weierstrass- i s cc hm- nh l Stone - Weierstrass

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3. nh l Ascoli- Tp ng lin tc- nh l Ascoli

13. Hng dn sinh vin t hcNi dung t hc:

- Chng 1: Cc minh ho cho cc khi nim chun, khng gian nh chun v mt skhng gian nh chun c th: C[a, b],Lp(X), lp- Chng 2: Cc v d minh ho cho cc nh l c bn ca gii tch hm- Chng 3: Cc minh ho cho cc khi nim dng Hermite, tch v hng, khng giantin Hilbert, khng gian HilbertHnh thc t hc:

- Ging vin trnh by s lc ni dung- Lp chia nhm t hc, vit bi thu hoch v tho lun

Trng Khoa Trng B mn Ngi lp

Nguyn Dng Hong Nguyn Vn Dng Nguyn Vn Dng

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Chng 0Kin thc chun b

0.1 i s tuyn tnh

Hng dn t hc1. Khng gian vect.

2. Khng gian vect con.

3. C s v chiu.

4. nh x tuyn tnh.

5. Khng gian thng.

0.2 Khng gian mtric v khng gian tpHng dn t hc

1. Khng gian mtric.

2. Khng gian tp.

3. Ln cn.

4. Tp ng.

5. Tnh lin tc.6. Khng gian tp con.

7. Tp tch.

8. Khng gian mtric y .

9. Khng gian compc.

0.3 Bi tpChun b cc bi tp trang 15-17.

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Chng 1Khng gian nh chun v khng gianBanach

1.1 Khng gian nh chun v khng gian Banach

Hng dn t hc1. Khi nim chun, khng gian nh chun, khng gian Banach v v d minh ho.

2. Chun l s suy rng ca khi nim no?

3. Mi quan h gia chun v mtric?

4. Tnh lin tc ca nh x chun v cc php ton trn khng gian nh chun.

5. Khng gian Eclide.

6. Khng gian C[a, b] vi chun hi t u v chun tch phn.

7. Khi nim tp li, tp b chn, on, khong trong khng gian vct v v d minhho.

8. Tnh cht ca c s li v b chn ca khng gian nh chun.

9. Bi tp 1, 2, 3.

1.2 Chui trong khng gian nh chunHng dn t hc1. Nhc li khi nim chui s trong R v cc tnh cht c bn ca chui s.

2. Khi nim chui trong khng gian nh chun, tng ring ca chui, chui hi t, chuiphn k trong khng gian nh chun v v d minh ho.

3. Chui l s suy rng ca khi nim no?

4. Tiu chun Cauchy v s hi t ca chui trong khng gian nh chun.

5. Tng ca hai chui v tch ca mt chui vi mt s.

6. Khi nim chui hi t tuyt i trong khng gian nh chun v v d minh ho.

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7. Mi quan h gia chui hi t v hi t tuyt i trong R v trong khng gian nhchun.

8. iu kin cn v ca khng gian Banach.

9. Bi tp 5, 6.

1.3 Khng gian Lp(X)

Hng dn t hc1. Bt ng thc Holder.

2. Bt ng thc Mincowski.

3. nh ngha khng gian Lp(X) v L1(X), L2(X).

4. Chun trong khng gian Lp(X).5. Khng gian Lp(X) c l khng gian Banach khng?

6. nh ngha khng gian lp v l1, l2.

7. Chun trong khng gian lp.

8. Khng gian lp c l khng gian Banach khng?

9. Mi quan h gia khng gian Lp(X) v khng gian lp.

1.4 nh x tuyn tnh lin tcHng dn t hc

1. Nhc li khi nim nh x tuyn tnh v phim hm tuyn tnh gia hai khng gianvect.

2. Mi quan h gia tnh lin tc u, tnh lin tc v tnh b chn ca nh x tuyn tnhlin tc.

3. Nhc li khi nim nh x ng cu gia hai khng gian vect (ng cu i s).

4. Nhc li khi nim nh x ng phi gia hai khng gian tp.

5. Khi nim nh x ng cu gia hai khng gian nh chun.

6. Hai khng gian nh th no th c gi l ng cu vi nhau?

7. iu kin cn v mt ng cu i s l ng cu.

8. Hai chun nh th no c gi l tng ng vi nhau? iu kin cn v haichun l tng ng vi nhau.

9. Tnh Banach ca khng gian nh chun c thay i khng khi thay chun xut phtbi chun tng ng?

10. Cc k hiu L(E, F),L(E, F), E#, E.

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11. Vi f L(E, F) th k hiu f c hiu nh th no?

12. Cc cng thc tnh ca f?

13. L(E, F) c l khng gian vect khng?

14. L(E, F) c l khng gian nh chun khng?

15. Trnh by v d v nh x tuyn tnh lin tc trn Kn v tnh chun ca nh x tuyntnh ?

16. Trnh by v d v nh x tuyn tnh lin tc trn C[a, b] v tnh chun ca nh xtuyn tnh ?

17. iu kin L(E, F) l khng gian Banach?

18. Hp thnh ca hai nh x tuyn tnh lin tc c l nh x tuyn tnh lin tc haykhng?

19. Khng gian nh chun F c nhng vo khng gian L(K, F) nh th no?

20. Khng gian lin hp E c Banach khng?

21. Bi tp 7, 8, 9, 10, 12.

1.5 Khng gian con v khng gian thng

Hng dn t hc

1. Nhc li khi nim khng gian vect con v tiu chun kim tra khng gian vectcon.

2. Nhc li mi quan h gia tnh ng v tnh y ca khng gian mtric.

3. Mi quan h gia tnh ng v tnh Banach ca khng gian nh chun.

4. Nu F l khng gian vect con th F c l khng gian vect con hay khng?

5. Nhc li khi nim tng trc tip ca hai khng gian vect v tiu chun kim tratng trc tip.

6. Cc php chiu p1, p2 c hiu nh th no? Cho v d minh ho.7. Tnh cht ca php chiu.

8. Tng trc tip tp c hiu nh th no? Cho v d minh ho.

9. Tnh cht ca tng trc tip tp.

10. Khi nim phn b tp. Cho v d minh ho.

11. Nhc li khi nim tch Descarte ca hai khng gian vect.

12. Tch Descarte ca hai khng gian vect c th l khng gian nh chun vi chun nhth no?

13. Cho v d v khng gian nh chun tch.

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14. Mi quan h gia tng trc tip v tch ca hai khng gian nh chun.

15. Khi nim siu phng trong khng gian vect (khng gian nh chun). Cho v dminh ho.

16. Tnh cht ng v tr mt ca siu phng trong khng gian nh chun.

17. Mi quan h gia siu phng v phim hm tuyn tnh.18. Khi nim phng trnh ca siu phng c hiu nh th no? Cho v d minh ho.

19. Mi quan h gia siu phng ng v phim hm tuyn tnh lin tc.

20. Nhc li khi nim tp thng.

21. Tp thng E/F ca khng gian nh chun E vi F l khng gian con ng ca Ec hiu nh th no?

22. E/F l khng gian nh chun vi chun nh th no?

23. Khi nim khng gian nh chun thng v v d minh ho.

24. Tnh Banach ca khng gian nh chun thng.

1.6 Khng gian chiu v khng gian kh li

Hng dn t hc1. c trng ca khng gian nh chun hu hn chiu.

2. Khng gian hu hn chiu c Banach khng? Cc chun trn khng gian hu hn chiuc tng ng vi nhau khng?

3. Tnh cht ng ca khng gian con ca khng gian nh chun hu hn chiu.

4. Tnh cht ca phim hm tuyn tnh v nh x tuyn tnh xc nh trn khng gianhu hn chiu.

5. Tnh cht ca tng khng gian vect con ng v khng gian con hu hn chiu.

6. Khi nim khng gian compc a phng.

7. nh l Riesz.

8. Khi nim tp ton vn, dy ton vn trong khng gian nh chun v v d minh ho.

9. c trng ca nh x tuyn tnh lin tc trn tp ton vn.

10. c trng ca nh x tuyn tnh lin tc trn khng gian con tr mt.

11. Nhc li khi nim khng gian kh li v v d minh ho.

12. iu kin cn v khng gian nh chun l khng gian kh li.

13. Bi tp 14, 16.

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Chng 2Cc nh l c bn ca gii tch hm

2.1 nh l Hahn-Banach

Hng dn t hc1. Nhc li khi nim quan h th t, quan h th t sp tuyn tnh, bin trn v phn

t cc i.

2. Nhc li B Zorn.

3. Khi nim s chun v v d minh ho.

4. Khi nim na chun v v d minh ho.

5. Mi quan h gia s chun v na chun.

6. Tnh cht ca na chun.7. nh l Hahn-Banach cho khng gian vect thc.

8. Khi nim phim hm tuyn tnh thc v v d minh ho.

9. Mi quan h gia phim hm tuyn tnh thc v phim hm tuyn tnh phc.

10. nh l Hahn-Banach cho khng gian vect phc.

11. Mt s h qu ca nh l Hahn-Banach.

12. Bi tp 1, 2.

2.2 nh l nh x m v nh l th ng

Hng dn t hc1. Nhc li khi nim khng u tr mt, khng gian thuc phm tr th nht, khng

gian thuc phm tr th hai.

2. Nhc li nh l Baire v phm tr.

3. Gi s f : E F l nh x tuyn tnh lin tc t khng gian Banach E ln khnggian Banach F. Khi nh ca hnh cu n v m trong E c tnh cht g?

4. nh l nh x m.

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5. iu kin cho song nh tuyn tnh lin tc l ng phi.

6. c trng ca cc chun lm cho khng gian l Banach.

7. Khi nim th ca nh x.

8. c trng th ca nh x lin tc.

9. nh l th ng.

10. Bi tp 4, 5, 6, 7.

2.3 Nguyn l b chn u

Hng dn t hc1. Tnh lin tc u, lin tc v b chn ca na chun lin tc.

2. Tnh cht ca h na chun lin tc trn khng gian Banach E.

3. Nguyn l b chn u.

4. nh l Banach-Steinhaus.

2.4 Khng gian cc hm

Hng dn t hc

1. K hiu BF(A).2. BF(A) l khng gian vect vi hai php ton nh th no?

3. BF(A) l khng gian nh chun vi chun nh th no?

4. iu kin BF(A) l khng gian Banach.

5. Cc k hiu CF (X), CF(X) v mi quan h gia chng.

6. Hi t theo chun v hi t theo im trong BK(A) c hiu nh th no v mi quanh gia chng l g?

7. Cc thut ng chun hi t u, bao ng u, ...trong BK(A).

8. Tnh lin tc ca gii hn ca dy hm hi t.

9. nh l Dini.

10. Bi tp 7.

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Chng 3Khng gian Hilbert

3.1 Tch v hng

Hng dn t hc1. Dng Hermite l g? Cho v d minh ho?

2. Dng Hermite l s suy rng ca khi nim no?

3. Nu l dng Hermite trn E v x, y l t hp tuyn tnh cc vect th (x, y) cbiu din nh th no?

4. Dng Hermite trn khng gian hu hn chiu c hon ton xc nh bi cc gi trno?

5. Dng Hermite dng l g? Cho v d minh ho?

6. Pht biu bt ng thc Cauchy-Schwartz v c bit ho vo trong R v R2.

7. Pht biu bt ng thc Mincowski v c bit ho vo trong R v R2.

8. Tch v hng l g? Cho v d minh ho.

9. Tnh lin tc ca tch v hng.

10. iu kin cn v mt dng Hermite dng l tch v hng.

11. Khng gian tin Hilbert l g? Tch v hng l g? Chun sinh bi tch v hng l

g? Cho v d minh ho?12. Khi nim hai vect trc giao v v d minh ho.

13. nh l Pythagore trong khng gian tin Hilbert.

14. nh x ng cu gia hai khng gian tin Hilbert v v d minh ho. Mi quan hgia ng cu tin Hilbert v ng cu nh chun.

15. Khi nim khng gian tin Hilbert con v khi nim khng gian Hilbert. Cho v dminh ho.

16. ng thc bnh hnh trong khng gian tin Hilbert.

17. Khng gian Euclide.

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18. Khng gian 2.

19. Khng gian L2(X).

3.2 H trc giao

Hng dn t hc1. Khi nim h trc giao v v d minh ho.

2. Mi quan h gia h trc giao v h c lp tuyn tnh.

3. Php trc giao ho Gram-Schmidt v cho v d minh ho.

4. Khi nim vect trc giao vi tp hp v hai tp hp trc giao vi nhau. Cho v dminh ho.

5. Khi nim phn b trc giao v v d minh ho.6. Tnh cht ng ca phn b trc giao.

7. Khi nim hnh chiu trc giao v cho v d minh ho.

8. Tnh cht ca phn b trc giao ca khng gian con Hilbert trong khng gian tinHilbert.

9. c trng ca phim hm tuyn tnh lin tc trong khng gian tin Hilbert. Cho vd minh ho.

10. iu kin cn v cho h trc giao l ton vn.

3.3 Tng Hilbert ca cc khng gian Hilbert

Hng dn t hc1. Tng Hilbert ca dy cc khng gian Hilbert {En}n c hiu nh th no. Cho v d

minh ho.

2. Tng Hilbert cc khng gian Hilbert c l khng gian Hilbert hay khng?

3.4 H trc chun

Hng dn t hc1. H trc chun c hiu nh th no. Cho v d minh ho.

2. Trc chun ho ca h trc giao c hiu nh th no? Cho v d minh ho.

3. Mt s tnh cht n gin ca h trc chun.

4. c trng ca khng gian Hilbert hu hn chiu. Lin h vi c trng ca khng giannh chun hu hn chiu .

5. c trng ca dy trc chun {ei}i trong khng gian Hilbert.

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6. Biu din chui Fourier v ng thc Parsenal trong khng gian Hilbert.

7. Cc iu kin cn v cho dy trc chun {ei}i l ton vn trong khng gian Hilbert.

8. c trng kh li ca khng gian Hilbert v hn chiu.

Bi tp 1, 2, 3, 4, 8.

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Chng 4L thuyt ton t

4.1 Khng gian lin hp

Hng dn t hc1. Khng gian lin hp v khng gian lin hp th hai ca khng gian nh chun E.

2. nh x x nhng chnh tc c hiu nh th no? Cho v d minh ho.

3. Khng gian nh chun E c nhng ng c vo E nh th no?

4. Tp yu c hiu nh th no? Cho v d minh ho.

5. c trng ca c s ca tp yu.

6. Hi t yu c hiu nh th no? Cho v d minh ho.

7. c trng ca hi t yu trong khng gian nh chun.

8. Khi nim tp yu trn E. Cho v d minh ho.

4.2 Ton t compc

Hng dn t hc1. Khi nim ton t compc v v d minh ho.

2. Cc iu kin tng ng ca ton t compc.3. Tnh compc ca nh x hp thnh.

4. H tt c cc ton t compc c l khng gian vect khng?

5. Gii hn ca mt dy cc ton t compc c compc khng?

4.3 Ton t b chn trong khng gian Hilbert

Hng dn t hc1. K hiu L(E) c hiu nh th no?

2. Khi nim ton t lin hp, k hiu v v d minh ho.

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3. c trng ca ton t lin hp.

4. Mt s tnh cht n gin ca ton t lin hp.

5. Khi nim ton t t lin hp v v d minh ho.

6. iu kin cn v cho ton t l t lin hp.

7. Khi nim ton t dng v v d minh ho.

8. Mi quan h gia ton t dng v ton t lin hp.

9. Bt ng thc Cauchy-Schwartz tng qut. Lin h vi bt ng thc Cauchy-Schwartz hc.

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Ti liu tham kho[1] u Th Cp, Gii tch hm, NXB Gio dc, 2002.

[2] Nguyn Vn Khu v L Mu Hi, C s l thuyt hm v gii tch hm, NXB Gio dc,2002.

huongdantuhoc-giaitichham-09-09-09

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