HomeHome Metamath Proof Explorer
Theorem List (p. 227 of 315)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21459)
  Hilbert Space Explorer  Hilbert Space Explorer
(21460-22982)
  Users' Mathboxes  Users' Mathboxes
(22983-31404)
 

Theorem List for Metamath Proof Explorer - 22601-22700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnlelshi 22601 The null space of a linear functional is a subspace. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
 |-  T  e.  LinFn   =>    |-  ( null `  T )  e.  SH
 
Theoremnlelchi 22602 The null space of a continuous linear functional is a closed subspace. Remark 3.8 of [Beran] p. 103. (Contributed by NM, 11-Feb-2006.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  T  e.  LinFn   &    |-  T  e.  ConFn   =>    |-  ( null `  T )  e.  CH
 
17.6.11  Riesz lemma
 
Theoremriesz3i 22603* A continuous linear functional can be expressed as an inner product. Existence part of Theorem 3.9 of [Beran] p. 104. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinFn   &    |-  T  e.  ConFn   =>    |- 
 E. w  e.  ~H  A. v  e.  ~H  ( T `  v )  =  ( v  .ih  w )
 
Theoremriesz4i 22604* A continuous linear functional can be expressed as an inner product. Uniqueness part of Theorem 3.9 of [Beran] p. 104. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinFn   &    |-  T  e.  ConFn   =>    |- 
 E! w  e.  ~H  A. v  e.  ~H  ( T `  v )  =  ( v  .ih  w )
 
Theoremriesz4 22605* A continuous linear functional can be expressed as an inner product. Uniqueness part of Theorem 3.9 of [Beran] p. 104. See riesz2 22607 for the bounded linear functional version. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  ( LinFn  i^i  ConFn )  ->  E! w  e.  ~H  A. v  e. 
 ~H  ( T `  v )  =  (
 v  .ih  w )
 )
 
Theoremriesz1 22606* Part 1 of the Riesz representation theorem for bounded linear functionals. A linear functional is bounded iff its value can be expressed as an inner product. Part of Theorem 17.3 of [Halmos] p. 31. For part 2, see riesz2 22607. For the continuous linear functional version, see riesz3i 22603 and riesz4 22605. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
 |-  ( T  e.  LinFn  ->  (
 ( normfn `  T )  e.  RR  <->  E. y  e.  ~H  A. x  e.  ~H  ( T `  x )  =  ( x  .ih  y
 ) ) )
 
Theoremriesz2 22607* Part 2 of the Riesz representation theorem for bounded linear functionals. The value of a bounded linear functional corresponds to a unique inner product. Part of Theorem 17.3 of [Halmos] p. 31. For part 1, see riesz1 22606. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  LinFn  /\  ( normfn `  T )  e.  RR )  ->  E! y  e.  ~H  A. x  e.  ~H  ( T `  x )  =  ( x  .ih  y ) )
 
17.6.12  Adjoints (cont.)
 
Theoremcnlnadjlem1 22608* Lemma for cnlnadji 22617 (Theorem 3.10 of [Beran] p. 104: every continuous linear operator has an adjoint). The value of the auxiliary functional  G. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T  e.  ConOp   &    |-  G  =  ( g  e.  ~H  |->  ( ( T `  g ) 
 .ih  y ) )   =>    |-  ( A  e.  ~H  ->  ( G `  A )  =  ( ( T `  A )  .ih  y ) )
 
Theoremcnlnadjlem2 22609* Lemma for cnlnadji 22617. 
G is a continuous linear functional. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T  e.  ConOp   &    |-  G  =  ( g  e.  ~H  |->  ( ( T `  g ) 
 .ih  y ) )   =>    |-  ( y  e.  ~H  ->  ( G  e.  LinFn  /\  G  e.  ConFn ) )
 
Theoremcnlnadjlem3 22610* Lemma for cnlnadji 22617. By riesz4 22605, 
B is the unique vector such that  ( T `  v )  .ih  y
)  =  ( v 
.ih  w ) for all  v. (Contributed by NM, 17-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T  e.  ConOp   &    |-  G  =  ( g  e.  ~H  |->  ( ( T `  g ) 
 .ih  y ) )   &    |-  B  =  ( iota_ w  e. 
 ~H A. v  e.  ~H  ( ( T `  v )  .ih  y )  =  ( v  .ih  w ) )   =>    |-  ( y  e.  ~H  ->  B  e.  ~H )
 
Theoremcnlnadjlem4 22611* Lemma for cnlnadji 22617. The values of auxiliary function  F are vectors. (Contributed by NM, 17-Feb-2006.) (Proof shortened by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T  e.  ConOp   &    |-  G  =  ( g  e.  ~H  |->  ( ( T `  g ) 
 .ih  y ) )   &    |-  B  =  ( iota_ w  e. 
 ~H A. v  e.  ~H  ( ( T `  v )  .ih  y )  =  ( v  .ih  w ) )   &    |-  F  =  ( y  e.  ~H  |->  B )   =>    |-  ( A  e.  ~H  ->  ( F `  A )  e.  ~H )
 
Theoremcnlnadjlem5 22612* Lemma for cnlnadji 22617. 
F is an adjoint of  T (later, we will show it is unique). (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T  e.  ConOp   &    |-  G  =  ( g  e.  ~H  |->  ( ( T `  g ) 
 .ih  y ) )   &    |-  B  =  ( iota_ w  e. 
 ~H A. v  e.  ~H  ( ( T `  v )  .ih  y )  =  ( v  .ih  w ) )   &    |-  F  =  ( y  e.  ~H  |->  B )   =>    |-  ( ( A  e.  ~H 
 /\  C  e.  ~H )  ->  ( ( T `
  C )  .ih  A )  =  ( C 
 .ih  ( F `  A ) ) )
 
Theoremcnlnadjlem6 22613* Lemma for cnlnadji 22617. 
F is linear. (Contributed by NM, 17-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T  e.  ConOp   &    |-  G  =  ( g  e.  ~H  |->  ( ( T `  g ) 
 .ih  y ) )   &    |-  B  =  ( iota_ w  e. 
 ~H A. v  e.  ~H  ( ( T `  v )  .ih  y )  =  ( v  .ih  w ) )   &    |-  F  =  ( y  e.  ~H  |->  B )   =>    |-  F  e.  LinOp
 
Theoremcnlnadjlem7 22614* Lemma for cnlnadji 22617. Helper lemma to show that  F is continuous. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T  e.  ConOp   &    |-  G  =  ( g  e.  ~H  |->  ( ( T `  g ) 
 .ih  y ) )   &    |-  B  =  ( iota_ w  e. 
 ~H A. v  e.  ~H  ( ( T `  v )  .ih  y )  =  ( v  .ih  w ) )   &    |-  F  =  ( y  e.  ~H  |->  B )   =>    |-  ( A  e.  ~H  ->  ( normh `  ( F `  A ) )  <_  ( ( normop `  T )  x.  ( normh `  A ) ) )
 
Theoremcnlnadjlem8 22615* Lemma for cnlnadji 22617. 
F is continuous. (Contributed by NM, 17-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T  e.  ConOp   &    |-  G  =  ( g  e.  ~H  |->  ( ( T `  g ) 
 .ih  y ) )   &    |-  B  =  ( iota_ w  e. 
 ~H A. v  e.  ~H  ( ( T `  v )  .ih  y )  =  ( v  .ih  w ) )   &    |-  F  =  ( y  e.  ~H  |->  B )   =>    |-  F  e.  ConOp
 
Theoremcnlnadjlem9 22616* Lemma for cnlnadji 22617. 
F provides an example showing the existence of a continuous linear adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T  e.  ConOp   &    |-  G  =  ( g  e.  ~H  |->  ( ( T `  g ) 
 .ih  y ) )   &    |-  B  =  ( iota_ w  e. 
 ~H A. v  e.  ~H  ( ( T `  v )  .ih  y )  =  ( v  .ih  w ) )   &    |-  F  =  ( y  e.  ~H  |->  B )   =>    |- 
 E. t  e.  ( LinOp  i^i  ConOp ) A. x  e.  ~H  A. z  e. 
 ~H  ( ( T `
  x )  .ih  z )  =  ( x  .ih  ( t `  z ) )
 
Theoremcnlnadji 22617* Every continuous linear operator has an adjoint. Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T  e.  ConOp   =>    |- 
 E. t  e.  ( LinOp  i^i  ConOp ) A. x  e.  ~H  A. y  e. 
 ~H  ( ( T `
  x )  .ih  y )  =  ( x  .ih  ( t `  y ) )
 
Theoremcnlnadjeui 22618* Every continuous linear operator has a unique adjoint. Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T  e.  ConOp   =>    |- 
 E! t  e.  ( LinOp  i^i  ConOp ) A. x  e.  ~H  A. y  e. 
 ~H  ( ( T `
  x )  .ih  y )  =  ( x  .ih  ( t `  y ) )
 
Theoremcnlnadjeu 22619* Every continuous linear operator has a unique adjoint. Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  ( LinOp  i^i  ConOp )  ->  E! t  e.  ( LinOp  i^i  ConOp ) A. x  e.  ~H  A. y  e.  ~H  ( ( T `
  x )  .ih  y )  =  ( x  .ih  ( t `  y ) ) )
 
Theoremcnlnadj 22620* Every continuous linear operator has an adjoint. Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  ( LinOp  i^i  ConOp )  ->  E. t  e.  ( LinOp  i^i  ConOp ) A. x  e.  ~H  A. y  e.  ~H  ( ( T `
  x )  .ih  y )  =  ( x  .ih  ( t `  y ) ) )
 
Theoremcnlnssadj 22621 Every continuous linear Hilbert space operator has an adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
 |-  ( LinOp  i^i  ConOp )  C_  dom  adjh
 
Theorembdopssadj 22622 Every bounded linear Hilbert space operator has an adjoint. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
 |-  BndLinOp  C_  dom  adjh
 
Theorembdopadj 22623 Every bounded linear Hilbert space operator has an adjoint. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  BndLinOp  ->  T  e.  dom  adjh )
 
Theoremadjbdln 22624 The adjoint of a bounded linear operator is a bounded linear operator. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  BndLinOp  ->  ( adjh `  T )  e.  BndLinOp )
 
Theoremadjbdlnb 22625 An operator is bounded and linear iff its adjoint is. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  BndLinOp  <->  ( adjh `  T )  e.  BndLinOp )
 
Theoremadjbd1o 22626 The mapping of adjoints of bounded linear operators is one-to-one onto. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
 |-  ( adjh 
 |`  BndLinOp ) : BndLinOp -1-1-onto->
 BndLinOp
 
Theoremadjlnop 22627 The adjoint of an operator is linear. Proposition 1 of [AkhiezerGlazman] p. 80. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.)
 |-  ( T  e.  dom  adjh  ->  (
 adjh `  T )  e. 
 LinOp )
 
Theoremadjsslnop 22628 Every operator with an adjoint is linear. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.)
 |-  dom  adjh  C_  LinOp
 
Theoremnmopadjlei 22629 Property of the norm of an adjoint. Part of proof of Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)
 |-  T  e. 
 BndLinOp   =>    |-  ( A  e.  ~H  ->  ( normh `  ( ( adjh `  T ) `  A ) )  <_  ( ( normop `  T )  x.  ( normh `  A ) ) )
 
Theoremnmopadjlem 22630 Lemma for nmopadji 22631. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)
 |-  T  e. 
 BndLinOp   =>    |-  ( normop `  ( adjh `  T ) )  <_  ( normop `  T )
 
Theoremnmopadji 22631 Property of the norm of an adjoint. Theorem 3.11(v) of [Beran] p. 106. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)
 |-  T  e. 
 BndLinOp   =>    |-  ( normop `  ( adjh `  T ) )  =  ( normop `  T )
 
Theoremadjeq0 22632 An operator is zero iff its adjoint is zero. Theorem 3.11(i) of [Beran] p. 106. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
 |-  ( T  =  0hop  <->  ( adjh `  T )  =  0hop )
 
Theoremadjmul 22633 The adjoint of the scalar product of an operator. Theorem 3.11(ii) of [Beran] p. 106. (Contributed by NM, 21-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T  e.  dom  adjh ) 
 ->  ( adjh `  ( A  .op  T ) )  =  ( ( * `  A )  .op  ( adjh `  T ) ) )
 
Theoremadjadd 22634 The adjoint of the sum of two operators. Theorem 3.11(iii) of [Beran] p. 106. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( S  e.  dom  adjh  /\  T  e.  dom  adjh ) 
 ->  ( adjh `  ( S  +op  T ) )  =  ( ( adjh `  S )  +op  ( adjh `  T ) ) )
 
Theoremnmoptrii 22635 Triangle inequality for the norms of bounded linear operators. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  S  e. 
 BndLinOp   &    |-  T  e.  BndLinOp   =>    |-  ( normop `  ( S  +op  T ) )  <_  ( ( normop `  S )  +  ( normop `  T ) )
 
Theoremnmopcoi 22636 Upper bound for the norm of the composition of two bounded linear operators. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  S  e. 
 BndLinOp   &    |-  T  e.  BndLinOp   =>    |-  ( normop `  ( S  o.  T ) )  <_  ( ( normop `  S )  x.  ( normop `  T ) )
 
Theorembdophsi 22637 The sum of two bounded linear operators is a bounded linear operator. (Contributed by NM, 9-Mar-2006.) (New usage is discouraged.)
 |-  S  e. 
 BndLinOp   &    |-  T  e.  BndLinOp   =>    |-  ( S  +op  T )  e.  BndLinOp
 
Theorembdophdi 22638 The difference between two bounded linear operators is bounded. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  S  e. 
 BndLinOp   &    |-  T  e.  BndLinOp   =>    |-  ( S  -op  T )  e.  BndLinOp
 
Theorembdopcoi 22639 The composition of two bounded linear operators is bounded. (Contributed by NM, 9-Mar-2006.) (New usage is discouraged.)
 |-  S  e. 
 BndLinOp   &    |-  T  e.  BndLinOp   =>    |-  ( S  o.  T )  e.  BndLinOp
 
Theoremnmoptri2i 22640 Triangle-type inequality for the norms of bounded linear operators. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  S  e. 
 BndLinOp   &    |-  T  e.  BndLinOp   =>    |-  ( ( normop `  S )  -  ( normop `  T ) )  <_  ( normop `  ( S  +op  T ) )
 
Theoremadjcoi 22641 The adjoint of a composition of bounded linear operators. Theorem 3.11(viii) of [Beran] p. 106. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  S  e. 
 BndLinOp   &    |-  T  e.  BndLinOp   =>    |-  ( adjh `  ( S  o.  T ) )  =  ( ( adjh `  T )  o.  ( adjh `  S ) )
 
Theoremnmopcoadji 22642 The norm of an operator composed with its adjoint. Part of Theorem 3.11(vi) of [Beran] p. 106. (Contributed by NM, 8-Mar-2006.) (New usage is discouraged.)
 |-  T  e. 
 BndLinOp   =>    |-  ( normop `  ( ( adjh `  T )  o.  T ) )  =  ( ( normop `  T ) ^ 2 )
 
Theoremnmopcoadj2i 22643 The norm of an operator composed with its adjoint. Part of Theorem 3.11(vi) of [Beran] p. 106. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  T  e. 
 BndLinOp   =>    |-  ( normop `  ( T  o.  ( adjh `  T )
 ) )  =  ( ( normop `  T ) ^ 2 )
 
Theoremnmopcoadj0i 22644 An operator composed with its adjoint is zero iff the operator is zero. Theorem 3.11(vii) of [Beran] p. 106. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  T  e. 
 BndLinOp   =>    |-  ( ( T  o.  ( adjh `  T )
 )  =  0hop  <->  T  =  0hop )
 
17.6.13  Quantum computation error bound theorem
 
Theoremunierri 22645 If we approximate a chain of unitary transformations (quantum computer gates)  F,  G by other unitary transformations  S,  T, the error increases at most additively. Equation 4.73 of [NielsenChuang] p. 195. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  F  e.  UniOp   &    |-  G  e.  UniOp   &    |-  S  e.  UniOp   &    |-  T  e.  UniOp   =>    |-  ( normop `  ( ( F  o.  G )  -op  ( S  o.  T ) ) )  <_  ( ( normop `  ( F  -op  S ) )  +  ( normop `  ( G  -op  T ) ) )
 
17.6.14  Dirac bra-ket notation (cont.)
 
Theorembranmfn 22646 The norm of the bra function. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  (
 normfn `  ( bra `  A ) )  =  ( normh `  A ) )
 
Theorembrabn 22647 The bra of a vector is a bounded functional. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  (
 normfn `  ( bra `  A ) )  e.  RR )
 
Theoremrnbra 22648 The set of bras equals the set of continuous linear functionals. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
 |-  ran  bra 
 =  ( LinFn  i^i  ConFn )
 
Theorembra11 22649 The bra function maps vectors one-to-one onto the set of continuous linear functionals. (Contributed by NM, 26-May-2006.) (Proof shortened by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  bra : ~H
 -1-1-onto-> ( LinFn  i^i  ConFn )
 
Theorembracnln 22650 A bra is a continuous linear functional. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( bra `  A )  e.  ( LinFn  i^i  ConFn ) )
 
Theoremcnvbraval 22651* Value of the converse of the bra function. Based on the Riesz Lemma riesz4 22605, this very important theorem not only justifies the Dirac bra-ket notation, but allows us to extract a unique vector from any continuous linear functional from which the functional can be recovered; i.e. a single vector can "store" all of the information contained in any entire continuous linear functional (mapping from  ~H to  CC). (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
 |-  ( T  e.  ( LinFn  i^i  ConFn )  ->  ( `' bra `  T )  =  ( iota_ y  e.  ~H A. x  e.  ~H  ( T `  x )  =  ( x  .ih  y
 ) ) )
 
Theoremcnvbracl 22652 Closure of the converse of the bra function. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
 |-  ( T  e.  ( LinFn  i^i  ConFn )  ->  ( `' bra `  T )  e. 
 ~H )
 
Theoremcnvbrabra 22653 The converse bra of the bra of a vector is the vector itself. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( `' bra `  ( bra `  A ) )  =  A )
 
Theorembracnvbra 22654 The bra of the converse bra of a continuous linear functional. (Contributed by NM, 31-May-2006.) (New usage is discouraged.)
 |-  ( T  e.  ( LinFn  i^i  ConFn )  ->  ( bra `  ( `' bra `  T ) )  =  T )
 
Theorembracnlnval 22655* The vector that a continuous linear functional is the bra of. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
 |-  ( T  e.  ( LinFn  i^i  ConFn )  ->  T  =  ( bra `  ( iota_ y  e. 
 ~H A. x  e.  ~H  ( T `  x )  =  ( x  .ih  y ) ) ) )
 
Theoremcnvbramul 22656 Multiplication property of the converse bra function. (Contributed by NM, 31-May-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T  e.  ( LinFn  i^i  ConFn ) )  ->  ( `' bra `  ( A  .fn  T ) )  =  ( ( * `  A )  .h  ( `' bra `  T )
 ) )
 
Theoremkbass1 22657 Dirac bra-ket associative law  (  |  A >.  <. B  |  )  |  C >.  =  |  A >. (
<. B  |  C >. ) i.e. the juxtaposition of an outer product with a ket equals a bra juxtaposed with an inner product. Since  <. B  |  C >. is a complex number, it is the first argument in the inner product  .h that it is mapped to, although in Dirac notation it is placed after the ket. (Contributed by NM, 15-May-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  ketbra  B ) `  C )  =  ( ( ( bra `  B ) `  C )  .h  A ) )
 
Theoremkbass2 22658 Dirac bra-ket associative law  ( <. A  |  B >. ) <. C  |  =  <. A  | 
(  |  B >.  <. C  |  ) i.e. the juxtaposition of an inner product with a bra equals a ket juxtaposed with an outer product. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( ( bra `  A ) `  B )  .fn  ( bra `  C ) )  =  (
 ( bra `  A )  o.  ( B  ketbra  C ) ) )
 
Theoremkbass3 22659 Dirac bra-ket associative law 
<. A  |  B >. 
<. C  |  D >.  =  ( <. A  |  B >.  <. C  |  )  |  D >.. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  ~H 
 /\  B  e.  ~H )  /\  ( C  e.  ~H 
 /\  D  e.  ~H ) )  ->  ( ( ( bra `  A ) `  B )  x.  ( ( bra `  C ) `  D ) )  =  ( ( ( ( bra `  A ) `  B )  .fn  ( bra `  C )
 ) `  D )
 )
 
Theoremkbass4 22660 Dirac bra-ket associative law 
<. A  |  B >. 
<. C  |  D >.  =  <. A  | 
(  |  B >.  <. C  |  D >. ). (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  ~H 
 /\  B  e.  ~H )  /\  ( C  e.  ~H 
 /\  D  e.  ~H ) )  ->  ( ( ( bra `  A ) `  B )  x.  ( ( bra `  C ) `  D ) )  =  ( ( bra `  A ) `  (
 ( ( bra `  C ) `  D )  .h  B ) ) )
 
Theoremkbass5 22661 Dirac bra-ket associative law  (  |  A >.  <. B  |  ) (  |  C >.  <. D  | 
)  =  ( (  |  A >.  <. B  | 
)  |  C >. )
<. D  |. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  ~H 
 /\  B  e.  ~H )  /\  ( C  e.  ~H 
 /\  D  e.  ~H ) )  ->  ( ( A  ketbra  B )  o.  ( C  ketbra  D ) )  =  ( ( ( A  ketbra  B ) `
  C )  ketbra  D ) )
 
Theoremkbass6 22662 Dirac bra-ket associative law  (  |  A >.  <. B  |  ) (  |  C >.  <. D  | 
)  =  |  A >.  ( <. B  |  (  |  C >.  <. D  | 
) ). (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  ~H 
 /\  B  e.  ~H )  /\  ( C  e.  ~H 
 /\  D  e.  ~H ) )  ->  ( ( A  ketbra  B )  o.  ( C  ketbra  D ) )  =  ( A 
 ketbra  ( `' bra `  (
 ( bra `  B )  o.  ( C  ketbra  D ) ) ) ) )
 
17.6.15  Positive operators (cont.)
 
Theoremleopg 22663* Ordering relation for positive operators. Definition of positive operator ordering in [Kreyszig] p. 470. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  A  /\  U  e.  B ) 
 ->  ( T  <_op  U  <->  ( ( U 
 -op  T )  e.  HrmOp  /\ 
 A. x  e.  ~H  0  <_  ( ( ( U  -op  T ) `
  x )  .ih  x ) ) ) )
 
Theoremleop 22664* Ordering relation for operators. Definition of positive operator ordering in [Kreyszig] p. 470. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  HrmOp  /\  U  e.  HrmOp )  ->  ( T  <_op  U  <->  A. x  e.  ~H  0  <_  ( ( ( U  -op  T ) `
  x )  .ih  x ) ) )
 
Theoremleop2 22665* Ordering relation for operators. Definition of operator ordering in [Young] p. 141. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  HrmOp  /\  U  e.  HrmOp )  ->  ( T  <_op  U  <->  A. x  e.  ~H  ( ( T `  x )  .ih  x ) 
 <_  ( ( U `  x )  .ih  x ) ) )
 
Theoremleop3 22666 Operator ordering in terms of a positive operator. Definition of operator ordering in [Retherford] p. 49. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  HrmOp  /\  U  e.  HrmOp )  ->  ( T  <_op  U  <->  0hop  <_op  ( U  -op  T ) ) )
 
Theoremleoppos 22667* Binary relation defining a positive operator. Definition VI.1 of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
 |-  ( T  e.  HrmOp  ->  ( 0hop  <_op  T  <->  A. x  e.  ~H  0  <_  ( ( T `
  x )  .ih  x ) ) )
 
Theoremleoprf2 22668 The ordering relation for operators is reflexive. (Contributed by NM, 24-Jul-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  T 
 <_op  T )
 
Theoremleoprf 22669 The ordering relation for operators is reflexive. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
 |-  ( T  e.  HrmOp  ->  T  <_op  T )
 
Theoremleopsq 22670 The square of a Hermitian operator is positive. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
 |-  ( T  e.  HrmOp  ->  0hop  <_op  ( T  o.  T ) )
 
Theorem0leop 22671 The zero operator is a positive operator. (The literature calls it "positive," even though in some sense it is really "nonnegative.") Part of Example 12.2(i) in [Young] p. 142. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
 |-  0hop  <_op  0hop
 
Theoremidleop 22672 The identity operator is a positive operator. Part of Example 12.2(i) in [Young] p. 142. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
 |-  0hop  <_op  Iop
 
Theoremleopadd 22673 The sum of two positive operators is positive. Exercise 1(i) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
 |-  (
 ( ( T  e.  HrmOp  /\  U  e.  HrmOp )  /\  ( 0hop  <_op  T  /\  0hop  <_op  U ) )  ->  0hop  <_op  ( T  +op  U ) )
 
Theoremleopmuli 22674 The scalar product of a nonnegative real and a positive operator is a positive operator. Exercise 1(ii) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  RR  /\  T  e.  HrmOp ) 
 /\  ( 0  <_  A  /\  0hop  <_op  T ) )  ->  0hop  <_op  ( A  .op  T ) )
 
Theoremleopmul 22675 The scalar product of a positive real and a positive operator is a positive operator. Exercise 1(ii) of [Retherford] p. 49. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  RR  /\  T  e.  HrmOp  /\  0  <  A )  ->  ( 0hop  <_op  T  <->  0hop  <_op  ( A  .op  T ) ) )
 
Theoremleopmul2i 22676 Scalar product applied to operator ordering. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  RR  /\  T  e.  HrmOp  /\  U  e.  HrmOp )  /\  ( 0  <_  A  /\  T  <_op  U )
 )  ->  ( A  .op  T )  <_op  ( A  .op  U ) )
 
Theoremleoptri 22677 The positive operator ordering relation satisfies trichotomy. Exercise 1(iii) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  HrmOp  /\  U  e.  HrmOp )  ->  ( ( T  <_op  U 
 /\  U  <_op  T )  <->  T  =  U )
 )
 
Theoremleoptr 22678 The positive operator ordering relation is transitive. Exercise 1(iv) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
 |-  (
 ( ( S  e.  HrmOp  /\  T  e.  HrmOp  /\  U  e.  HrmOp )  /\  ( S  <_op  T  /\  T  <_op  U ) )  ->  S  <_op  U )
 
Theoremleopnmid 22679 A bounded Hermitian operator is less than or equal to its norm times the identity operator. (Contributed by NM, 11-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR )  ->  T  <_op  ( ( normop `  T )  .op  Iop  ) )
 
Theoremnmopleid 22680 A nonzero, bounded Hermitian operator divided by its norm is less than or equal to the identity operator. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  HrmOp  /\  ( normop `  T )  e.  RR  /\  T  =/=  0hop
 )  ->  ( (
 1  /  ( normop `  T ) )  .op  T ) 
 <_op  Iop  )
 
Theoremopsqrlem1 22681* Lemma for opsqri . (Contributed by NM, 9-Aug-2006.) (New usage is discouraged.)
 |-  T  e.  HrmOp   &    |-  ( normop `  T )  e.  RR   &    |-  0hop  <_op  T   &    |-  R  =  ( ( 1  /  ( normop `  T )
 )  .op  T )   &    |-  ( T  =/=  0hop  ->  E. u  e.  HrmOp  ( 0hop  <_op  u  /\  ( u  o.  u )  =  R )
 )   =>    |-  ( T  =/=  0hop  ->  E. v  e.  HrmOp  ( 0hop  <_op 
 v  /\  ( v  o.  v )  =  T ) )
 
Theoremopsqrlem2 22682* Lemma for opsqri .  F `  N is the recursive function An (starting at n=1 instead of 0) of Theorem 9.4-2 of [Kreyszig] p. 476. (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
 |-  T  e.  HrmOp   &    |-  S  =  ( x  e.  HrmOp ,  y  e. 
 HrmOp  |->  ( x  +op  ( ( 1  / 
 2 )  .op  ( T  -op  ( x  o.  x ) ) ) ) )   &    |-  F  =  seq  1 ( S ,  ( NN  X.  { 0hop } ) )   =>    |-  ( F `  1
 )  =  0hop
 
Theoremopsqrlem3 22683* Lemma for opsqri . (Contributed by NM, 22-Aug-2006.) (New usage is discouraged.)
 |-  T  e.  HrmOp   &    |-  S  =  ( x  e.  HrmOp ,  y  e. 
 HrmOp  |->  ( x  +op  ( ( 1  / 
 2 )  .op  ( T  -op  ( x  o.  x ) ) ) ) )   &    |-  F  =  seq  1 ( S ,  ( NN  X.  { 0hop } ) )   =>    |-  ( ( G  e.  HrmOp  /\  H  e.  HrmOp )  ->  ( G S H )  =  ( G  +op  ( ( 1  / 
 2 )  .op  ( T  -op  ( G  o.  G ) ) ) ) )
 
Theoremopsqrlem4 22684* Lemma for opsqri . (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
 |-  T  e.  HrmOp   &    |-  S  =  ( x  e.  HrmOp ,  y  e. 
 HrmOp  |->  ( x  +op  ( ( 1  / 
 2 )  .op  ( T  -op  ( x  o.  x ) ) ) ) )   &    |-  F  =  seq  1 ( S ,  ( NN  X.  { 0hop } ) )   =>    |-  F : NN --> HrmOp
 
Theoremopsqrlem5 22685* Lemma for opsqri . (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
 |-  T  e.  HrmOp   &    |-  S  =  ( x  e.  HrmOp ,  y  e. 
 HrmOp  |->  ( x  +op  ( ( 1  / 
 2 )  .op  ( T  -op  ( x  o.  x ) ) ) ) )   &    |-  F  =  seq  1 ( S ,  ( NN  X.  { 0hop } ) )   =>    |-  ( N  e.  NN  ->  ( F `  ( N  +  1 )
 )  =  ( ( F `  N ) 
 +op  ( ( 1 
 /  2 )  .op  ( T  -op  ( ( F `  N )  o.  ( F `  N ) ) ) ) ) )
 
Theoremopsqrlem6 22686* Lemma for opsqri . (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
 |-  T  e.  HrmOp   &    |-  S  =  ( x  e.  HrmOp ,  y  e. 
 HrmOp  |->  ( x  +op  ( ( 1  / 
 2 )  .op  ( T  -op  ( x  o.  x ) ) ) ) )   &    |-  F  =  seq  1 ( S ,  ( NN  X.  { 0hop } ) )   &    |-  T  <_op  Iop   =>    |-  ( N  e.  NN  ->  ( F `  N )  <_op  Iop  )
 
17.6.16  Projectors as operators
 
Theorempjhmopi 22687 A projector is a Hermitian operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( proj  h `  H )  e.  HrmOp
 
Theorempjlnopi 22688 A projector is a linear operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( proj  h `  H )  e.  LinOp
 
Theorempjnmopi 22689 The operator norm of a projector on a nonzero closed subspace is one. Part of Theorem 26.1 of [Halmos] p. 43. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( H  =/=  0H  ->  ( normop `  ( proj  h `
  H ) )  =  1 )
 
Theorempjbdlni 22690 A projector is a bounded linear operator. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( proj  h `  H )  e.  BndLinOp
 
Theorempjhmop 22691 A projection is a Hermitian operator. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
 |-  ( H  e.  CH  ->  ( proj  h `  H )  e.  HrmOp )
 
Theoremhmopidmchi 22692 An idempotent Hermitian operator generates a closed subspace. Part of proof of Theorem of [AkhiezerGlazman] p. 64. (Contributed by NM, 21-Apr-2006.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  T  e.  HrmOp   &    |-  ( T  o.  T )  =  T   =>    |- 
 ran  T  e.  CH
 
Theoremhmopidmpji 22693 An idempotent Hermitian operator is a projection operator. Theorem 26.4 of [Halmos] p. 44. (Halmos seems to omit the proof that  H is a closed subspace, which is not trivial as hmopidmchi 22692 shows.) (Contributed by NM, 22-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  T  e.  HrmOp   &    |-  ( T  o.  T )  =  T   =>    |-  T  =  ( proj  h `
  ran  T )
 
Theoremhmopidmch 22694 An idempotent Hermitian operator generates a closed subspace. Part of proof of Theorem of [AkhiezerGlazman] p. 64. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  HrmOp  /\  ( T  o.  T )  =  T )  ->  ran  T  e.  CH )
 
Theoremhmopidmpj 22695 An idempotent Hermitian operator is a projection operator. Theorem 26.4 of [Halmos] p. 44. (Contributed by NM, 22-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  HrmOp  /\  ( T  o.  T )  =  T )  ->  T  =  ( proj  h `
  ran  T )
 )
 
Theorempjsdii 22696 Distributive law for Hilbert space operator sum. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( ( proj  h `
  H )  o.  ( S  +op  T ) )  =  (
 ( ( proj  h `  H )  o.  S )  +op  ( ( proj  h `
  H )  o.  T ) )
 
Theorempjddii 22697 Distributive law for Hilbert space operator difference. (Contributed by NM, 24-Nov-2000.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( ( proj  h `
  H )  o.  ( S  -op  T ) )  =  (
 ( ( proj  h `  H )  o.  S )  -op  ( ( proj  h `
  H )  o.  T ) )
 
Theorempjsdi2i 22698 Chained distributive law for Hilbert space operator difference. (Contributed by NM, 30-Nov-2000.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  R : ~H --> ~H   &    |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( ( R  o.  ( S  +op  T ) )  =  ( ( R  o.  S ) 
 +op  ( R  o.  T ) )  ->  ( ( ( proj  h `
  H )  o.  R )  o.  ( S  +op  T ) )  =  ( ( ( ( proj  h `  H )  o.  R )  o.  S )  +op  (
 ( ( proj  h `  H )  o.  R )  o.  T ) ) )
 
Theorempjcoi 22699 Composition of projections. (Contributed by NM, 16-Aug-2000.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( A  e.  ~H  ->  ( ( ( proj  h `  G )  o.  ( proj  h `  H ) ) `  A )  =  ( ( proj  h `
  G ) `  ( ( proj  h `  H ) `  A ) ) )
 
Theorempjcocli 22700 Closure of composition of projections. (Contributed by NM, 29-Nov-2000.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( A  e.  ~H  ->  ( ( ( proj  h `  G )  o.  ( proj  h `  H ) ) `  A )  e.  G )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31404
  Copyright terms: Public domain < Previous  Next >