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Theorem List for Metamath Proof Explorer - 21401-21500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremminvecolem3 21401* Lemma for minveco 21409. The sequence formed by taking elements successively closer to the infimum is Cauchy. (Contributed by Mario Carneiro, 8-May-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  ( normCV `  U )   &    |-  Y  =  ( BaseSet `  W )   &    |-  ( ph  ->  U  e.  CPreHil OLD )   &    |-  ( ph  ->  W  e.  ( ( SubSp `  U )  i^i  CBan ) )   &    |-  ( ph  ->  A  e.  X )   &    |-  D  =  ( IndMet `  U )   &    |-  J  =  ( MetOpen `  D )   &    |-  R  =  ran  (  y  e.  Y  |->  ( N `  ( A M y ) ) )   &    |-  S  =  sup ( R ,  RR ,  `'  <  )   &    |-  ( ph  ->  F : NN --> Y )   &    |-  ( ( ph  /\  n  e.  NN )  ->  (
 ( A D ( F `  n ) ) ^ 2 ) 
 <_  ( ( S ^
 2 )  +  (
 1  /  n )
 ) )   =>    |-  ( ph  ->  F  e.  ( Cau `  D ) )
 
Theoremminvecolem4a 21402* Lemma for minveco 21409. 
F is convergent in the subspace topology on  Y. (Contributed by Mario Carneiro, 7-May-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  ( normCV `  U )   &    |-  Y  =  ( BaseSet `  W )   &    |-  ( ph  ->  U  e.  CPreHil OLD )   &    |-  ( ph  ->  W  e.  ( ( SubSp `  U )  i^i  CBan ) )   &    |-  ( ph  ->  A  e.  X )   &    |-  D  =  ( IndMet `  U )   &    |-  J  =  ( MetOpen `  D )   &    |-  R  =  ran  (  y  e.  Y  |->  ( N `  ( A M y ) ) )   &    |-  S  =  sup ( R ,  RR ,  `'  <  )   &    |-  ( ph  ->  F : NN --> Y )   &    |-  ( ( ph  /\  n  e.  NN )  ->  (
 ( A D ( F `  n ) ) ^ 2 ) 
 <_  ( ( S ^
 2 )  +  (
 1  /  n )
 ) )   =>    |-  ( ph  ->  F (
 ~~> t `  ( MetOpen `  ( D  |`  ( Y  X.  Y ) ) ) ) ( ( ~~> t `  ( MetOpen `  ( D  |`  ( Y  X.  Y ) ) ) ) `  F ) )
 
Theoremminvecolem4b 21403* Lemma for minveco 21409. The convergent point of the cauchy sequence  F is a member of the base space. (Contributed by Mario Carneiro, 16-Jun-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  ( normCV `  U )   &    |-  Y  =  ( BaseSet `  W )   &    |-  ( ph  ->  U  e.  CPreHil OLD )   &    |-  ( ph  ->  W  e.  ( ( SubSp `  U )  i^i  CBan ) )   &    |-  ( ph  ->  A  e.  X )   &    |-  D  =  ( IndMet `  U )   &    |-  J  =  ( MetOpen `  D )   &    |-  R  =  ran  (  y  e.  Y  |->  ( N `  ( A M y ) ) )   &    |-  S  =  sup ( R ,  RR ,  `'  <  )   &    |-  ( ph  ->  F : NN --> Y )   &    |-  ( ( ph  /\  n  e.  NN )  ->  (
 ( A D ( F `  n ) ) ^ 2 ) 
 <_  ( ( S ^
 2 )  +  (
 1  /  n )
 ) )   =>    |-  ( ph  ->  (
 ( ~~> t `  J ) `  F )  e.  X )
 
Theoremminvecolem4c 21404* Lemma for minveco 21409. The infimum of the distances to  A is a real number. (Contributed by Mario Carneiro, 16-Jun-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  ( normCV `  U )   &    |-  Y  =  ( BaseSet `  W )   &    |-  ( ph  ->  U  e.  CPreHil OLD )   &    |-  ( ph  ->  W  e.  ( ( SubSp `  U )  i^i  CBan ) )   &    |-  ( ph  ->  A  e.  X )   &    |-  D  =  ( IndMet `  U )   &    |-  J  =  ( MetOpen `  D )   &    |-  R  =  ran  (  y  e.  Y  |->  ( N `  ( A M y ) ) )   &    |-  S  =  sup ( R ,  RR ,  `'  <  )   &    |-  ( ph  ->  F : NN --> Y )   &    |-  ( ( ph  /\  n  e.  NN )  ->  (
 ( A D ( F `  n ) ) ^ 2 ) 
 <_  ( ( S ^
 2 )  +  (
 1  /  n )
 ) )   =>    |-  ( ph  ->  S  e.  RR )
 
Theoremminvecolem4 21405* Lemma for minveco 21409. The convergent point of the cauchy sequence  F attains the minimum distance, and so is closer to  A than any other point in  Y. (Contributed by Mario Carneiro, 7-May-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  ( normCV `  U )   &    |-  Y  =  ( BaseSet `  W )   &    |-  ( ph  ->  U  e.  CPreHil OLD )   &    |-  ( ph  ->  W  e.  ( ( SubSp `  U )  i^i  CBan ) )   &    |-  ( ph  ->  A  e.  X )   &    |-  D  =  ( IndMet `  U )   &    |-  J  =  ( MetOpen `  D )   &    |-  R  =  ran  (  y  e.  Y  |->  ( N `  ( A M y ) ) )   &    |-  S  =  sup ( R ,  RR ,  `'  <  )   &    |-  ( ph  ->  F : NN --> Y )   &    |-  ( ( ph  /\  n  e.  NN )  ->  (
 ( A D ( F `  n ) ) ^ 2 ) 
 <_  ( ( S ^
 2 )  +  (
 1  /  n )
 ) )   &    |-  T  =  ( 1  /  ( ( ( ( ( A D ( ( ~~> t `  J ) `  F ) )  +  S )  /  2 ) ^
 2 )  -  ( S ^ 2 ) ) )   =>    |-  ( ph  ->  E. x  e.  Y  A. y  e.  Y  ( N `  ( A M x ) )  <_  ( N `  ( A M y ) ) )
 
Theoremminvecolem5 21406* Lemma for minveco 21409. Discharge the assumption about the sequence  F by applying countable choice ax-cc 8015. (Contributed by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  ( normCV `  U )   &    |-  Y  =  ( BaseSet `  W )   &    |-  ( ph  ->  U  e.  CPreHil OLD )   &    |-  ( ph  ->  W  e.  ( ( SubSp `  U )  i^i  CBan ) )   &    |-  ( ph  ->  A  e.  X )   &    |-  D  =  ( IndMet `  U )   &    |-  J  =  ( MetOpen `  D )   &    |-  R  =  ran  (  y  e.  Y  |->  ( N `  ( A M y ) ) )   &    |-  S  =  sup ( R ,  RR ,  `'  <  )   =>    |-  ( ph  ->  E. x  e.  Y  A. y  e.  Y  ( N `  ( A M x ) )  <_  ( N `  ( A M y ) ) )
 
Theoremminvecolem6 21407* Lemma for minveco 21409. Any minimal point is less than  S away from  A. (Contributed by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  ( normCV `  U )   &    |-  Y  =  ( BaseSet `  W )   &    |-  ( ph  ->  U  e.  CPreHil OLD )   &    |-  ( ph  ->  W  e.  ( ( SubSp `  U )  i^i  CBan ) )   &    |-  ( ph  ->  A  e.  X )   &    |-  D  =  ( IndMet `  U )   &    |-  J  =  ( MetOpen `  D )   &    |-  R  =  ran  (  y  e.  Y  |->  ( N `  ( A M y ) ) )   &    |-  S  =  sup ( R ,  RR ,  `'  <  )   =>    |-  ( ( ph  /\  x  e.  Y )  ->  (
 ( ( A D x ) ^ 2
 )  <_  ( ( S ^ 2 )  +  0 )  <->  A. y  e.  Y  ( N `  ( A M x ) ) 
 <_  ( N `  ( A M y ) ) ) )
 
Theoremminvecolem7 21408* Lemma for minveco 21409. Since any two minimal points are distance zero away from each other, the minimal point is unique. (Contributed by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  ( normCV `  U )   &    |-  Y  =  ( BaseSet `  W )   &    |-  ( ph  ->  U  e.  CPreHil OLD )   &    |-  ( ph  ->  W  e.  ( ( SubSp `  U )  i^i  CBan ) )   &    |-  ( ph  ->  A  e.  X )   &    |-  D  =  ( IndMet `  U )   &    |-  J  =  ( MetOpen `  D )   &    |-  R  =  ran  (  y  e.  Y  |->  ( N `  ( A M y ) ) )   &    |-  S  =  sup ( R ,  RR ,  `'  <  )   =>    |-  ( ph  ->  E! x  e.  Y  A. y  e.  Y  ( N `  ( A M x ) )  <_  ( N `  ( A M y ) ) )
 
Theoremminveco 21409* Minimizing vector theorem, or the Hilbert projection theorem. There is exactly one vector in a complete subspace  W that minimizes the distance to an arbitrary vector  A in a parent inner product space. Theorem 3.3-1 of [Kreyszig] p. 144, specialized to subspaces instead of convex subsets. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  ( normCV `  U )   &    |-  Y  =  ( BaseSet `  W )   &    |-  ( ph  ->  U  e.  CPreHil OLD )   &    |-  ( ph  ->  W  e.  ( ( SubSp `  U )  i^i  CBan ) )   &    |-  ( ph  ->  A  e.  X )   =>    |-  ( ph  ->  E! x  e.  Y  A. y  e.  Y  ( N `  ( A M x ) )  <_  ( N `  ( A M y ) ) )
 
15.8  Complex Hilbert spaces
 
15.8.1  Definition and basic properties
 
Syntaxchlo 21410 Extend class notation with the class of all complex Hilbert spaces.
 class  CHil OLD
 
Definitiondf-hlo 21411 Define the class of all complex Hilbert spaces. A Hilbert space is a Banach space which is also an inner product space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
 |- 
 CHil OLD  =  ( CBan  i^i  CPreHil
 OLD )
 
Theoremishlo 21412 The predicate "is a complex Hilbert space." A Hilbert space is a Banach space which is also an inner product space, i.e. whose norm satisfies the parallelogram law. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
 |-  ( U  e.  CHil OLD  <->  ( U  e.  CBan  /\  U  e. 
 CPreHil OLD ) )
 
Theoremhlobn 21413 Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
 |-  ( U  e.  CHil OLD 
 ->  U  e.  CBan )
 
Theoremhlph 21414 Every complex Hilbert space is an inner product space (also called a pre-Hilbert space). (Contributed by NM, 28-Apr-2007.) (New usage is discouraged.)
 |-  ( U  e.  CHil OLD 
 ->  U  e.  CPreHil OLD )
 
Theoremhlrel 21415 The class of all complex Hilbert spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
 |- 
 Rel  CHil OLD
 
Theoremhlnv 21416 Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
 |-  ( U  e.  CHil OLD 
 ->  U  e.  NrmCVec )
 
Theoremhlnvi 21417 Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
 |-  U  e.  CHil OLD   =>    |-  U  e.  NrmCVec
 
Theoremhlvc 21418 Every complex Hilbert space is a complex vector space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  W  =  ( 1st `  U )   =>    |-  ( U  e.  CHil OLD 
 ->  W  e.  CVec OLD )
 
Theoremhlcmet 21419 The induced metric on a complex Hilbert space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  D  =  (
 IndMet `  U )   =>    |-  ( U  e.  CHil OLD  ->  D  e.  ( CMet `  X ) )
 
Theoremhlmet 21420 The induced metric on a complex Hilbert space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  D  =  (
 IndMet `  U )   =>    |-  ( U  e.  CHil OLD  ->  D  e.  ( Met `  X ) )
 
Theoremhlpar2 21421 The parallelogram law satified by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  (
 normCV `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( ( ( N `
  ( A G B ) ) ^
 2 )  +  (
 ( N `  ( A M B ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `
  A ) ^
 2 )  +  (
 ( N `  B ) ^ 2 ) ) ) )
 
Theoremhlpar 21422 The parallelogram law satified by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  A  e.  X  /\  B  e.  X )  ->  ( ( ( N `  ( A G B ) ) ^ 2 )  +  ( ( N `  ( A G ( -u 1 S B ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `  A ) ^ 2 )  +  ( ( N `  B ) ^ 2
 ) ) ) )
 
15.8.2  Standard axioms for a complex Hilbert space
 
Theoremhlex 21423 The base set of a Hilbert space is a set. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   =>    |-  X  e.  _V
 
Theoremhladdf 21424 Mapping for Hilbert space vector addition. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( U  e.  CHil OLD  ->  G : ( X  X.  X ) --> X )
 
Theoremhlcom 21425 Hilbert space vector addition is commutative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( A G B )  =  ( B G A ) )
 
Theoremhlass 21426 Hilbert space vector addition is associative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( ( A G B ) G C )  =  ( A G ( B G C ) ) )
 
Theoremhl0cl 21427 The Hilbert space zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   =>    |-  ( U  e.  CHil OLD  ->  Z  e.  X )
 
Theoremhladdid 21428 Hilbert space addition with the zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  A  e.  X )  ->  ( A G Z )  =  A )
 
Theoremhlmulf 21429 Mapping for Hilbert space scalar multiplication. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( U  e.  CHil OLD 
 ->  S : ( CC 
 X.  X ) --> X )
 
Theoremhlmulid 21430 Hilbert space scalar multiplication by one. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  A  e.  X )  ->  ( 1 S A )  =  A )
 
Theoremhlmulass 21431 Hilbert space scalar multiplication associative law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X ) )  ->  ( ( A  x.  B ) S C )  =  ( A S ( B S C ) ) )
 
Theoremhldi 21432 Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  ( A S ( B G C ) )  =  ( ( A S B ) G ( A S C ) ) )
 
Theoremhldir 21433 Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X ) )  ->  ( ( A  +  B ) S C )  =  ( ( A S C ) G ( B S C ) ) )
 
Theoremhlmul0 21434 Hilbert space scalar multiplication by zero. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  A  e.  X )  ->  ( 0 S A )  =  Z )
 
Theoremhlipf 21435 Mapping for Hilbert space inner product. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( U  e.  CHil OLD 
 ->  P : ( X  X.  X ) --> CC )
 
Theoremhlipcj 21436 Conjugate law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( A P B )  =  ( * `  ( B P A ) ) )
 
Theoremhlipdir 21437 Distributive law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( ( A G B ) P C )  =  ( ( A P C )  +  ( B P C ) ) )
 
Theoremhlipass 21438 Associative law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i
 OLD `  U )   =>    |-  (
 ( U  e.  CHil OLD  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A S B ) P C )  =  ( A  x.  ( B P C ) ) )
 
Theoremhlipgt0 21439 The inner product of a Hilbert space vector by itself is positive. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  A  e.  X  /\  A  =/=  Z ) 
 ->  0  <  ( A P A ) )
 
Theoremhlcompl 21440 Completeness of a Hilbert space. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
 |-  D  =  ( IndMet `  U )   &    |-  J  =  (
 MetOpen `  D )   =>    |-  ( ( U  e.  CHil OLD  /\  F  e.  ( Cau `  D )
 )  ->  F  e.  dom  ( ~~> t `  J ) )
 
15.8.3  Examples of complex Hilbert spaces
 
Theoremcnchl 21441 The set of complex numbers is a complex Hilbert space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +  ,  x.  >. ,  abs >.   =>    |-  U  e.  CHil OLD
 
15.8.4  Subspaces
 
Theoremssphl 21442 A Banach subspace of an inner product space is a Hilbert space. (Contributed by NM, 11-Apr-2008.) (New usage is discouraged.)
 |-  H  =  ( SubSp `  U )   =>    |-  ( ( U  e.  CPreHil OLD  /\  W  e.  H  /\  W  e.  CBan )  ->  W  e.  CHil OLD )
 
15.8.5  Hellinger-Toeplitz Theorem
 
Theoremhtthlem 21443* Lemma for htth 21444. The collection  K, which consists of functions  F ( z ) ( w )  =  <. w  |  T
( z ) >.  =  <. T ( w )  |  z >. for each  z in the unit ball, is a collection of bounded linear functions by ipblnfi 21380, so by the Uniform Boundedness theorem ubth 21398, there is a uniform bound  y on  ||  F ( x )  || for all  x in the unit ball. Then  |  T (
x )  |  ^
2  =  <. T ( x )  |  T
( x ) >.  =  F ( x ) (  T ( x ) )  <_  y  |  T ( x )  |, so  |  T ( x )  |  <_  y and 
T is bounded. (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  L  =  ( U 
 LnOp  U )   &    |-  B  =  ( U  BLnOp  U )   &    |-  N  =  ( normCV `  U )   &    |-  U  e.  CHil OLD   &    |-  W  =  <. <.  +  ,  x.  >. ,  abs >.   &    |-  ( ph  ->  T  e.  L )   &    |-  ( ph  ->  A. x  e.  X  A. y  e.  X  ( x P ( T `  y
 ) )  =  ( ( T `  x ) P y ) )   &    |-  F  =  ( z  e.  X  |->  ( w  e.  X  |->  ( w P ( T `  z
 ) ) ) )   &    |-  K  =  ( F " { z  e.  X  |  ( N `  z
 )  <_  1 }
 )   =>    |-  ( ph  ->  T  e.  B )
 
Theoremhtth 21444* Hellinger-Toeplitz Theorem: any self-adjoint linear operator defined on all of Hilbert space is bounded. Theorem 10.1-1 of [Kreyszig] p. 525. Discovered by E. Hellinger and O. Toeplitz in 1910, "it aroused both admiration and puzzlement since the theorem establishes a relation between properties of two different kinds, namely, the properties of being defined everywhere and being bounded." (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  L  =  ( U 
 LnOp  U )   &    |-  B  =  ( U  BLnOp  U )   =>    |-  ( ( U  e.  CHil OLD  /\  T  e.  L  /\  A. x  e.  X  A. y  e.  X  ( x P ( T `  y
 ) )  =  ( ( T `  x ) P y ) ) 
 ->  T  e.  B )
 
15.9  Hilbert Space Explorer
 
15.9.1  Basic Hilbert space definitions
 
Syntaxchil 21445 Extend class notation with Hilbert vector space.
 class  ~H
 
Syntaxcva 21446 Extend class notation with vector addition in Hilbert space. In the literature, the subscript "v" is omitted, but we need it to avoid ambiguity with complex number addition  + caddc 8694.
 class  +h
 
Syntaxcsm 21447 Extend class notation with scalar multiplication in Hilbert space. In the literature scalar multiplication is usually indicated by juxtaposition, but we need an explicit symbol to prevent ambiguity.
 class  .h
 
Syntaxcsp 21448 Extend class notation with inner (scalar) product in Hilbert space. In the literature, the inner product of  A and  B is usually written  <. A ,  B >. but our operation notation allows us to use existing theorems about operations and also eliminates ambiguity with the definition of an ordered pair df-op 3609.
 class  .ih
 
Syntaxcno 21449 Extend class notation with the norm function in Hilbert space. In the literature, the norm of  A is usually written "||  A ||", but we use function notation to take advantage of our existing theorems about functions.
 class  normh
 
Syntaxc0v 21450 Extend class notation with zero vector in Hilbert space.
 class  0h
 
Syntaxcmv 21451 Extend class notation with vector subtraction in Hilbert space.
 class  -h
 
Syntaxccau 21452 Extend class notation with set of Cauchy sequences in Hilbert space.
 class  Cauchy
 
Syntaxchli 21453 Extend class notation with convergence relation in Hilbert space.
 class  ~~>v
 
Syntaxcsh 21454 Extend class notation with set of subspaces of a Hilbert space.
 class  SH
 
Syntaxcch 21455 Extend class notation with set of closed subspaces of a Hilbert space.
 class  CH
 
Syntaxcort 21456 Extend class notation with orthogonal complement in  CH.
 class  _|_
 
Syntaxcph 21457 Extend class notation with subspace sum in  CH.
 class  +H
 
Syntaxcspn 21458 Extend class notation with subspace span in  CH.
 class  span
 
Syntaxchj 21459 Extend class notation with join in  CH.
 class  vH
 
Syntaxchsup 21460 Extend class notation with supremum of a collection in  CH.
 class  \/H
 
Syntaxc0h 21461 Extend class notation with zero of  CH.
 class  0H
 
Syntaxccm 21462 Extend class notation with the commutes relation on a Hilbert lattice.
 class  C_H
 
Syntaxcpjh 21463 Extend class notation with set of projections on a Hilbert space.
 class  proj  h
 
Syntaxchos 21464 Extend class notation with sum of Hilbert space operators.
 class  +op
 
Syntaxchot 21465 Extend class notation with scalar product of a Hilbert space operator.
 class  .op
 
Syntaxchod 21466 Extend class notation with difference of Hilbert space operators.
 class  -op
 
Syntaxchfs 21467 Extend class notation with sum of Hilbert space functionals.
 class  +fn
 
Syntaxchft 21468 Extend class notation with scalar product of Hilbert space functional.
 class  .fn
 
Syntaxch0o 21469 Extend class notation with the Hilbert space zero operator.
 class  0hop
 
Syntaxchio 21470 Extend class notation with Hilbert space identity operator.
 class  Iop
 
Syntaxcnop 21471 Extend class notation with the operator norm function.
 class  normop
 
Syntaxccop 21472 Extend class notation with set of continuous Hilbert space operators.
 class  ConOp
 
Syntaxclo 21473 Extend class notation with set of linear Hilbert space operators.
 class  LinOp
 
Syntaxcbo 21474 Extend class notation with set of bounded linear operators.
 class  BndLinOp
 
Syntaxcuo 21475 Extend class notation with set of unitary Hilbert space operators.
 class  UniOp
 
Syntaxcho 21476 Extend class notation with set of Hermitian Hilbert space operators.
 class  HrmOp
 
Syntaxcnmf 21477 Extend class notation with the functional norm function.
 class  normfn
 
Syntaxcnl 21478 Extend class notation with the functional nullspace function.
 class  null
 
Syntaxccnfn 21479 Extend class notation with set of continuous Hilbert space functionals.
 class  ConFn
 
Syntaxclf 21480 Extend class notation with set of linear Hilbert space functionals.
 class  LinFn
 
Syntaxcado 21481 Extend class notation with Hilbert space adjoint function.
 class  adjh
 
Syntaxcbr 21482 Extend class notation with the bra of a vector in Dirac bra-ket notation.
 class  bra
 
Syntaxck 21483 Extend class notation with the outer product of two vectors in Dirac bra-ket notation.
 class  ketbra
 
Syntaxcleo 21484 Extend class notation with positive operator ordering.
 class  <_op
 
Syntaxcei 21485 Extend class notation with Hilbert space eigenvector function.
 class  eigvec
 
Syntaxcel 21486 Extend class notation with Hilbert space eigenvalue function.
 class  eigval
 
Syntaxcspc 21487 Extend class notation with the spectrum of an operator.
 class  Lambda
 
Syntaxcst 21488 Extend class notation with set of states on a Hilbert lattice.
 class  States
 
Syntaxchst 21489 Extend class notation with set of Hilbert-space-valued states on a Hilbert lattice.
 class  CHStates
 
Syntaxccv 21490 Extend class notation with the covers relation on a Hilbert lattice.
 class  <oH
 
Syntaxcat 21491 Extend class notation with set of atoms on a Hilbert lattice.
 class HAtoms
 
Syntaxcmd 21492 Extend class notation with the modular pair relation on a Hilbert lattice.
 class  MH
 
Syntaxcdmd 21493 Extend class notation with the dual modular pair relation on a Hilbert lattice.
 class  MH*
 
15.9.2  Preliminary ZFC lemmas
 
Definitiondf-hnorm 21494 Define the function for the norm of a vector of Hilbert space. See normval 21649 for its value and normcl 21650 for its closure. Theorems norm-i-i 21658, norm-ii-i 21662, and norm-iii-i 21664 show it has the expected properties of a norm. In the literature, the norm of  A is usually written "||  A ||", but we use function notation to take advantage of our existing theorems about functions. Definition of norm in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
 |-  normh  =  ( x  e.  dom  dom  .ih  |->  ( sqr `  ( x  .ih  x ) ) )
 
Definitiondf-hba 21495 Define base set of Hilbert space, for use if we want to develop Hilbert space independently from the axioms (see comments in ax-hilex 21525). Note that  ~H is considered a primitive in the Hilbert space axioms below, and we don't use this definition outside of this section. This definition can be proved independently from those axioms as as theorem hhba 21692. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
 
Definitiondf-h0v 21496 Define the zero vector of Hilbert space. Note that  0vec is considered a primitive in the Hilbert space axioms below, and we don't use this definition outside of this section. It is proved from the axioms as theorem hh0v 21693. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  0h  =  ( 0vec `  <. <.  +h  ,  .h  >. ,  normh >. )
 
Definitiondf-hvsub 21497* Define vector subtraction. See hvsubvali 21546 for its value and hvsubcli 21547 for its closure. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
 |-  -h  =  ( x  e.  ~H ,  y  e.  ~H  |->  ( x  +h  ( -u 1  .h  y ) ) )
 
Definitiondf-hlim 21498* Define the limit relation for Hilbert space. See hlimi 21713 for its relational expression. Note that  f : NN --> ~H is an infinite sequence of vectors, i.e. a mapping from integers to vectors. Definition of converge in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
 |-  ~~>v  =  { <. f ,  w >.  |  ( ( f : NN --> ~H  /\  w  e. 
 ~H )  /\  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
 ( normh `  ( (
 f `  z )  -h  w ) )  < 
 x ) }
 
Definitiondf-hcau 21499* Define the set of Cauchy sequences on a Hilbert space. See hcau 21709 for its membership relation. Note that  f : NN --> ~H is an infinite sequence of vectors, i.e. a mapping from integers to vectors. Definition of Cauchy sequence in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
 |-  Cauchy  =  {
 f  e.  ( ~H 
 ^m  NN )  |  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
 ( normh `  ( (
 f `  y )  -h  ( f `  z
 ) ) )  < 
 x }
 
Theoremh2hva 21500 The group (addition) operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  NrmCVec   =>    |- 
 +h  =  ( +v
 `  U )
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