함수해석학(2016_1)수업정리(방학때 옮길 것)

Chapter 1 Hilbert Space

*용어정의

1. vector space의 semi-inner product란? inner product란?

2. Hilbert space의 정의

3. Absolutely continuous for f:[a,b]->F의 정의와 동치 2가지

4. G:open in C일 때 the bergman space for G란?

5. perp(E)의 정의, projection of a closed linear subspace(P_K)의 정의

6. S:subset of HS, VS란?

6. bdd linear functional f:HS->F의 정의

7. basis for HS의 정의

8. infinite subset S of HS가 linearly independent란, any finite subset of S가 linearly independent

9. dimension of HS란?

10. HS1,HS2:isomorphic이란 linear, surjective, preserve ip map이 존재

11. f:MetricS1->MetricS2가 isometry란?

12. f:HS->HS가 unitary operator란?(isomorphism인데 정의역=공역)

13. direct sum of HS1,HS2란?direct sum of countable HS_i란? direct sum of HS_i란?(net)

 

*Thm

1. CBS inequality and equality iff there are scalar a,b(not both 0)...

2. Absolutely continuous for f:[a,b]->F의 성질

-+,-,*,/에 닫혀있다.

-Lipschitz Continous이면 Absolutely Continuous이다.

-Absolutely Continuous이면 Uniformly Continuous이다.

3. Completion of IPS(extention of <,>, norm, and metric)

4. Polar Identity란

5. if K:closed convex, then dist(h_o,K)=d(h_0,k_0)인 k_0 in K존재 and unique

6. if K:closed linear subspace, then dist(h_0,K)=d(h_0,k_0)인 k_0 in K존재 and unique and h_0 - k_0:orthogonal to K

7. if K:closed linear subspace and h_0 - k_0:orthogonal to K인 k_0 in K 존재, then dist(h_0,k_0)=d(h_0,k_0)

8. perp(E):closed linear subspace of HS

9. projection of a closed linear subspace(P_K)의 성질

-linear

-norm<=1

-idempotent

-ran(P_K)=K, ker(P_K)=perp(K)

10. K:closed linear subspace of HS이면 perp(perp(K))=K

11. perp(perp(E))=closed linear span of E

12. S:linear subspace of HS일 때 S:dense in HS iff perp(S)={0}

13. f:HS->F, linear, TFAE

-conti, uniformly conti, conti at 0, conti at any pt, bdd

14. (Riesz for HS)f:HS->F, linear, bdd일 떄 f(h)=<h,h_0>인 h_0가 유일하게 존재 with ||f||=||h_0||

15. (Gram-schmidt) S:linearly ind이면 S':orthonormal s.t. ....

16. S:finite orthonormal일때 projection of VS의 representation

17. (Bessel)S:countable orthonormal일때 sum of (coefficients)^2 <= ||h|| 

18. S:orthonormal일때 sum of (coefficients)^2 <= ||h||(net)

19. net으로 cv이면 걍 cv

20. S:orthonormal일때 sum of <h,s>s는 always cv(net)

21. S:orthonormal일때 TFAE

-S:basis

-hㅗS이면 h=0

-VS=HS

-h=sum of <h,s>s(net)

-<h1,h2>=sum of <h1,s><s,h2>(net)

-(Parseval's Identity)||h||=sum of ||<h,s>||^2(net)

22. S1,S2:basis then, |S1|=|S2|

23. HS:separable iff dim(HS):at most countable

24. f:HS1->HS2가 linear일때 f:isometry iff f:preserve ip

25. f:HS1->HS2가 linear, isometry일 때

-ran(f):closed

26. HS1,HS2:isomorphic iff dim(HS1)=dim(HS2)

-HS with basis S, HS와 l^2(S)는 isomorphic

(Fourier는 따로 보고, 보고나서는 Chapter2, Example 4.12보기)

 

Chapter 2 Operators on Hilbert Space

*용어정의

1. B(HS1,HS2)란?

2. M_Φ란?

3. k:MSxMS->F, measurable, K:integral operator란?

4. u:HS1xHS2->F, sesquilinear란? sesquilinear가 bdd란? 대표적인 bdd sesquilinear는?

5. f in B(HS1,HS2)일 때, adj(f)란?

6. f in B(HS1,HS2)가 invertible이란?

7. f in B(HS)가 hermitian이란?, normal이란?

8. f in B(HS)가 idempotent란?, projection이란?(of a closed linear space란 거 없이)

9. K_i:closed linear subspace of HS일 때 the direct sum of K_i란?

10. f in B(HS)에 대해 invariant subspace for f란?(closed필요), reducing subspace for f 란?(closed필요)

11. f in B(HS)에 대해 f:compact란?

12. B_0(HS1,HS2)란?

13. f:HS->HS가 finite rank를 가진다란?

14. B_00(HS1,HS2)란?

 

*Thm

1. f:HS1->HS2, linear, TFAE

-conti, uniformly conti, conti at 0, conti at any pt, bdd

2. B(HS1,HS2):NVS

3. f in B(HS1,HS2), g in B(HS2,HS3), then g o f in B(HS1,HS3)

4. u:sesquilinear, bdd이면 u(h1,h2)=<Ah1,h2>=<h1,Bh2>인 A:HS1->HS2, B:HS2->HS1, linear, bdd가 유일하게 존재

5. f in B(HS1,HS2)일때 f:surjective isometry iff adj(f)=inv(f)

6. about adjoint, f,g in B(HS), a:scalar

-adj(af+g)=conj(a)adj(f)+adj(g)

-adj(gf)=adj(f)adj(g)

-adj(adj(f))=f

-if f:invertible, then adj(f):invertible and adj(inv(f))=inv(adj(f))

-||f||=||adj(f)||=(||adj(f) o f||)^1/2

7. (Consequence of Open Mapping Theorem) f in B(HS1,HS2), bijective이면 invertible이다.

8. f:HS->HS, linear, bdd일때 f:hermitian iff <f(h),h>:real for all h in HS(Base field가 C일 때만 성립)

9. f:HS->HS, hermitian일때

-||f||:=sup over ||h||=1 {|<f(h),h>|}

-<f(h),h>=0 for all h in HS이면 f=0(HS over C이면 f:hermitian조건 없어도 됨)

10. f:HS->HS, linear, bdd일때 TFAE

-f:normal

-||f(h)||=||adj(f)(h)|| for all h

-the real and imaginary parts of f commute(HS over C일 때만)

(real part of f := (f+adj(f))/2, imaginary part of f := (f-adj(f))/2

11. f:HS->HS, linear, bdd일때 TFAE

-f:isometry

-f:preserve ip

-adj(f)f=identity

12. f:HS->HS, linear bdd일 때 TFAE

-adj(f)f=fadj(f)=identity

-f:unitary

-f:normal isometry

13. f:HS->HS, linear bdd일 때

ker(f)=perp(ran(adj(f)))

ker(adj(f))=perp(ran(f))

perp(ker(f))=closure of ran(adj(f))

perp(ker(adj(f))= closure of ran(f)

(perp(ker(f))=ran(adj(f))는 아닐 수 있음, 조심)

14. f:idempotent iff I-f:idempotent

15. if f:idempotent, then ran(I-f)=ker(f), ker(I-f)=ran(f), and HS=the direct sum of ker(f) and ran(f) (여기서 direct sum은 그냥 Vector Space에서의 direct sum)

16. HS=the direct sum of S1 and S2이면 te! f in B(HS) s.t. ran(f)=S1 and ker(f)=S2

17. f:nonzero idempotent이면 TFAE

-f:projection

-f:orthogonal projection onto ran(f)

-||f||=1

-f:hermitian

-f:normal

-<f(h),h)> >= 0 for all h

18. K1,K2:closed linear subspace of HS일때 the direct sum of K1 and K2는 K1+K2와 같다.(finite는 다 됨, infinite는 안됨)

19. K:closed linear subspace of HS일 때 HS=the direct sum of K and perp(K)

20. f in B(HS), K:closed linear subspace of HS, P:projection onto K, TFAE

-K:invariant subspace for f

-PfP=fP

-f:K->perp(K)는 0가 된다.

21. f in B(HS), K:closed linear subspace of HS, P:projection onto K, TFAE

-K:reducing subspace for f

-Pf=fP

-f:K->perp(K)는 0, f:perp(K)->K도 0

-K:reducing subsapce for f and adj(f)

22. f in B(HS), K:invariant closed linear subspace of HS일 때 ||the restriction of f onto K||<=||f||

23. B_0(HS1,HS2):closed in B(HS1,HS2)

24. f in B(HS1), g in B(HS2), T in B_0(HS1,HS2)일 때 Tf in B_0(HS1,HS2)이고 gT in B_0(HS1,HS2)이다.(two-sided ideal)

25. f in B(HS1,HS2)일때 TFAE

-f:compact

-adj(f):compact

-te seq {f_n} s.t. f_n:finite rank and ||f_n - f||->0

26. f in B_0(HS1,HS2)일 때

-cl(ran(f)):separable

-if {e_n}:basis for cl(ran(f)) and P_n:projection onto V{e_j | 1<= j <=n}, then ||P_n o f - f||->0

27. HS:separable with basis {e_n} and {α_n} s.t. sup|α_n|=M<inf일때

-if f e_n=α_n e_n for all n, then f can be extended by linearity to a bdd f with ||f||=M. 그리고 그 f:compact iff α_n -> 0

28. f in B_0(HS), λ:nonzero egv for f일때, eigenspace는 finite dimension

29. f in B_0(HS), λ:nonzero s.t. inf{||(f-λ)h|| over ||h||=1}=0이면 λ:egv for f

30. f in B_0(HS), λ:nonzero, nonegv for f, conj(λ):nonegv for adj(f)이면 ran(f-λ)=HS and inv(f-λ):bdd

 

Chapter 3 Banach Spaces

*용어정의

1. seminorm on VS란?

2. two norms가 equivalent란?

3. C_b(X)란?C_0(X)란?C^(n)[0,1]이란?

4. NVS1,NVS2가 isometrically isomorphic이란?(linear, surjective, isometry)(그냥 isomorphic하면, linear, bijective, homeo를 가리킴, 즉 topologically iso)

5. direct sum of NVS using finite p, using inf, using cv to

6. hyplerplane of VS란?

7. NVS가 reflexive란?

 

 

*Thm

1. two norms가 equivalent iff C1||x||<=|||x|||<=C2||x||

2. B(NVS1,NVS2):BS iff NVS2:BS

3. dim(NVS):finite이면 any two norms on NVS are equivalent

4. any finite dimensional linear manifold is closed in larger one.

5. f:NVS1->NVS2, linear, dim(NVS1):finite이면 f:continuous

6. K:closed linear subspace of NVS이면

-NVS/K:normed vector space

-natural map Q:NVS->NVS/K, linear, conti, open

-if NVS:BS, then NVS/K도 BS

-U:open in NVS iff Q(U):open in NVS/K

7. X:NVS, M:closed linear subspace of X, N:finite dim of X이면 M+N은 closed linear subspace of X

8. direct sum of NVS using cv to 0 is linear subspace of direct sum of NVS using inf

9. X:direct sum of NVS using finite p

-X:NVS and projection onto NVS_i is linear, conti, bdd with 1 norm, open map, surjective

-X:BS iff NVS_i:BS

10. S:hyperplane of NVS이면 S는 dense in NVS or closed in NVS

11. f:NVS->F, linear functional일때 f:conti iff ker(f):closed

12. (NVS)^*:BS(if NVS is nonzero)

13. (Hahn-Banach Theorem)X:VS over R and g:sunlinear functional, S:linear subspace of X, f:linear functional on S s.t. f<=g

then te F:X->R s.t. F=f on S and F<=g

(이 Theorem의 의미는 g에 dominate가 계속 유지되면서 extension을 찾는 것임)

(Lemma6.3부터 Cor6.8까지 읽기)

(Thm 6.13부터 Cor6.14까지 읽기)

14. (Open Mapping Theorem)f in B(BS1,BS2)가 surjective이면 open map

15. (Inverse Mapping Theorem)f in B(BS1,BS2)가 bijective이면 f^(-1)도 conti(bdd)

16. (Closed Graph Theorem)f:BS1->BS2, linear가 closed graph를 가지면 f는 conti

17. (Uniform Boundedness Principle)F:a collection of f:BS->NVS s.t. for any x in BS, sup over f {|f(x)|}<inf이면 sup over f ||f||<inf

 

*example

1. semi-inner product인데 not inner product인 예?

2. {f:[0,1]->F, f:AC, f(0)=0, f' in L^2[0,1]} with int from x=0 to x=1 f'coj(g')=<f,g>, HS임을 보여라.

3. 걍 cv인데 not net cv인 예?(net cv를 abs cv로 생각하면 예 찾기 쉬움)

4. idempotent인데 not projection인 예 (1 0 1 0) matrix

5. NVS/S가 seminorm은 되는데 norm안되는 예 X=C_0, M=C_00

6. natural map Q:NVS->NVS/K가 not closed map인걸 보이는 예, X=L^2(-pi,pi), M=V{e_n}, F=V{f_n} where f_n(t) = e_(-n)(t) + n*e_n(t), e_n(t)=exp(int)

7. ker(linear functional):not closed in NVS, dense in NVS인 예 X=C_0(N), {e_n(i)=1 for only i=n}, x_0(i)=1/i, then Hamel basis containing {e_1,...,x_0}, make f

 

*특정 concepts

-L^p with MS, 1<=p<=inf

-BS

-(L^p)^* = L^q

-p=1일땐 sf-M여야 가능

-reflexive for 1<p<inf

-about K with k

-K:L^p(MS)->L^p(MS), linear, bdd, ||K|| <= (c_1)^1/p * (c_2)^1/q

-l^2(I)

-{simple functions}:dense in l^2(I)

-HS

-basis = {e_i s.t. i in I}

-HS with basis S일때 HS와 l^2(S)는 isormophic as HS

-I=N일 때

-unilateral shift

-isometry

-not surjective

-not normal

-backward shift

-adj(unilateral shift)

-l^inf(I)

-I=N일때

-all bdd seq of scalars

-the bergman space for G

-HS(Cauchy, Riesz, Morera Theorems 필요)

-L^2[0,2pi] with C

-HS

-basis = {1/sqrt(2pi) * exp(int) s.t. n in Z}

-L^2(-pi,pi) with C

 

-L^2 with MS

-about K with k

-K:L^2(MS)->L^2(MS), linear, bdd, ||K||<=(c1*c2)^(1/2), compact operator, ||K|| <= ||k||_2 (L^2 norm)

-adj(K)는 with kernel conj(k(y,x))

-about Volterra, k:[0,1]x[0,1] ->R, char function of {(x,y) s.t. y<x}

-no eigenvalues

-k:L^2(-pi,pi)xL^2(-pi,pi) -> C,

-L^2 with sf-M

-M_Φ in BS(L^2) and ||M_Φ||=||Φ||

-adj(M_Φ)=M_(conj(Φ))

-M_Φ:normal

-M_Φ:hermitian iff Φ:real

-M_Φ:unitary iff |Φ|=1 a.e.

-L^p with sf-M, 1 <= p <= inf

-M_Φ in BS(L^p) with ||M_Φ||=||Φ||

-C_b(X)

-BS

-C_0(X)

-closed in C_b(X)

-X=N일때

-all seq cv to 0

-C^(n)[0,1]

-BS