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Theorem List for Metamath Proof Explorer - 22201-22300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsiilem2 22201 Lemma for sii 22203. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  N  =  (
 normCV `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   &    |-  M  =  ( -v `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( C  e.  CC  /\  ( C  x.  ( A P B ) )  e. 
 RR  /\  0  <_  ( C  x.  ( A P B ) ) )  ->  ( ( B P A )  =  ( C  x.  (
 ( N `  B ) ^ 2 ) ) 
 ->  ( sqr `  (
 ( A P B )  x.  ( C  x.  ( ( N `  B ) ^ 2
 ) ) ) ) 
 <_  ( ( N `  A )  x.  ( N `  B ) ) ) )
 
Theoremsiii 22202 Inference from sii 22203. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  N  =  (
 normCV `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   =>    |-  ( abs `  ( A P B ) ) 
 <_  ( ( N `  A )  x.  ( N `  B ) )
 
Theoremsii 22203 Schwarz inequality. Part of Lemma 3-2.1(a) of [Kreyszig] p. 137. This is also called the Cauchy-Schwarz inequality by some authors and Bunjakovaskij-Cauchy-Schwarz inequality by others. See also theorems bcseqi 22470, bcsiALT 22529, bcsiHIL 22530, csbrn 26147. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  N  =  (
 normCV `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   =>    |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( abs `  ( A P B ) ) 
 <_  ( ( N `  A )  x.  ( N `  B ) ) )
 
Theoremsspph 22204 A subspace of an inner product space is an inner product space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
 |-  H  =  ( SubSp `  U )   =>    |-  ( ( U  e.  CPreHil OLD  /\  W  e.  H ) 
 ->  W  e.  CPreHil OLD )
 
Theoremipblnfi 22205* A function  F generated by varying the first argument of an inner product (with its second argument a fixed vector  A) is a bounded linear functional, i.e. a bounded linear operator from the vector space to  CC. (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  C  =  <. <.  +  ,  x.  >. ,  abs >.   &    |-  B  =  ( U  BLnOp  C )   &    |-  F  =  ( x  e.  X  |->  ( x P A ) )   =>    |-  ( A  e.  X  ->  F  e.  B )
 
Theoremip2eqi 22206* Two vectors are equal iff their inner products with all other vectors are equal. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   =>    |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A. x  e.  X  ( x P A )  =  ( x P B )  <->  A  =  B )
 )
 
Theoremphoeqi 22207* A condition implying that two operators are equal. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   =>    |-  ( ( S : Y
 --> X  /\  T : Y
 --> X )  ->  ( A. x  e.  X  A. y  e.  Y  ( x P ( S `
  y ) )  =  ( x P ( T `  y
 ) )  <->  S  =  T ) )
 
Theoremajmoi 22208* Every operator has at most one adjoint. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   =>    |- 
 E* s ( s : Y --> X  /\  A. x  e.  X  A. y  e.  Y  (
 ( T `  x ) Q y )  =  ( x P ( s `  y ) ) )
 
Theoremajfuni 22209 The adjoint function is a function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
 |-  A  =  ( U adj W )   &    |-  U  e. 
 CPreHil OLD   &    |-  W  e.  NrmCVec   =>    |- 
 Fun  A
 
Theoremajfun 22210 The adjoint function is a function. This is not immediately apparent from df-aj 22099 but results from the uniqueness shown by ajmoi 22208. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
 |-  A  =  ( U adj W )   =>    |-  ( ( U  e.  CPreHil OLD  /\  W  e.  NrmCVec )  ->  Fun  A )
 
Theoremajval 22211* Value of the adjoint function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  P  =  ( .i OLD `  U )   &    |-  Q  =  ( .i
 OLD `  W )   &    |-  A  =  ( U adj W )   =>    |-  ( ( U  e.  CPreHil OLD  /\  W  e.  NrmCVec  /\  T : X --> Y )  ->  ( A `  T )  =  ( iota s
 ( s : Y --> X  /\  A. x  e.  X  A. y  e.  Y  ( ( T `
  x ) Q y )  =  ( x P ( s `
  y ) ) ) ) )
 
17.5  Complex Banach spaces
 
17.5.1  Definition and basic properties
 
Syntaxccbn 22212 Extend class notation with the class of all complex Banach spaces.
 class  CBan
 
Definitiondf-cbn 22213 Define the class of all complex Banach spaces. (Contributed by NM, 5-Dec-2006.) (New usage is discouraged.)
 |- 
 CBan  =  { u  e. 
 NrmCVec  |  ( IndMet `  u )  e.  ( CMet `  ( BaseSet `  u )
 ) }
 
Theoremiscbn 22214 A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  D  =  (
 IndMet `  U )   =>    |-  ( U  e.  CBan  <->  ( U  e.  NrmCVec  /\  D  e.  ( CMet `  X )
 ) )
 
Theoremcbncms 22215 The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  D  =  (
 IndMet `  U )   =>    |-  ( U  e.  CBan 
 ->  D  e.  ( CMet `  X ) )
 
Theorembnnv 22216 Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
 |-  ( U  e.  CBan  ->  U  e.  NrmCVec )
 
Theorembnrel 22217 The class of all complex Banach spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
 |- 
 Rel  CBan
 
Theorembnsscmcl 22218 A subspace of a Banach space is a Banach space iff it is closed in the norm-induced metric of the parent space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  D  =  (
 IndMet `  U )   &    |-  J  =  ( MetOpen `  D )   &    |-  H  =  ( SubSp `  U )   &    |-  Y  =  ( BaseSet `  W )   =>    |-  (
 ( U  e.  CBan  /\  W  e.  H ) 
 ->  ( W  e.  CBan  <->  Y  e.  ( Clsd `  J )
 ) )
 
17.5.2  Examples of complex Banach spaces
 
Theoremcnbn 22219 The set of complex numbers is a complex Banach space. (Contributed by Steve Rodriguez, 4-Jan-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +  ,  x.  >. ,  abs >.   =>    |-  U  e.  CBan
 
17.5.3  Uniform Boundedness Theorem
 
Theoremubthlem1 22220* Lemma for ubth 22223. The function  A exhibits a countable collection of sets that are closed, being the inverse image under  t of the closed ball of radius  k, and by assumption they cover  X. Thus, by the Baire Category theorem bcth2 19152, for some  n the set  A `  n has an interior, meaning that there is a closed ball  { z  e.  X  |  ( y D z )  <_  r } in the set. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  N  =  (
 normCV `  W )   &    |-  D  =  (
 IndMet `  U )   &    |-  J  =  ( MetOpen `  D )   &    |-  U  e.  CBan   &    |-  W  e.  NrmCVec   &    |-  ( ph  ->  T  C_  ( U  BLnOp  W ) )   &    |-  ( ph  ->  A. x  e.  X  E. c  e.  RR  A. t  e.  T  ( N `  ( t `  x ) )  <_  c )   &    |-  A  =  ( k  e.  NN  |->  { z  e.  X  |  A. t  e.  T  ( N `  ( t `
  z ) ) 
 <_  k } )   =>    |-  ( ph  ->  E. n  e.  NN  E. y  e.  X  E. r  e.  RR+  { z  e.  X  |  ( y D z )  <_  r }  C_  ( A `
  n ) )
 
Theoremubthlem2 22221* Lemma for ubth 22223. Given that there is a closed ball  B ( P ,  R ) in  A `  K, for any  x  e.  B
( 0 ,  1 ), we have  P  +  R  x.  x  e.  B
( P ,  R
) and  P  e.  B
( P ,  R
), so both of these have 
norm ( t ( z ) )  <_  K and so  norm ( t ( x  ) )  <_ 
( norm ( t ( P ) )  + 
norm ( t ( P  +  R  x.  x ) ) )  /  R  <_  (  K  +  K
)  /  R, which is our desired uniform bound. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  N  =  (
 normCV `  W )   &    |-  D  =  (
 IndMet `  U )   &    |-  J  =  ( MetOpen `  D )   &    |-  U  e.  CBan   &    |-  W  e.  NrmCVec   &    |-  ( ph  ->  T  C_  ( U  BLnOp  W ) )   &    |-  ( ph  ->  A. x  e.  X  E. c  e.  RR  A. t  e.  T  ( N `  ( t `  x ) )  <_  c )   &    |-  A  =  ( k  e.  NN  |->  { z  e.  X  |  A. t  e.  T  ( N `  ( t `
  z ) ) 
 <_  k } )   &    |-  ( ph  ->  K  e.  NN )   &    |-  ( ph  ->  P  e.  X )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  { z  e.  X  |  ( P D z ) 
 <_  R }  C_  ( A `  K ) )   =>    |-  ( ph  ->  E. d  e.  RR  A. t  e.  T  ( ( U
 normOp OLD W ) `  t )  <_  d )
 
Theoremubthlem3 22222* Lemma for ubth 22223. Prove the reverse implication, using nmblolbi 22149. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  N  =  (
 normCV `  W )   &    |-  D  =  (
 IndMet `  U )   &    |-  J  =  ( MetOpen `  D )   &    |-  U  e.  CBan   &    |-  W  e.  NrmCVec   &    |-  ( ph  ->  T  C_  ( U  BLnOp  W ) )   =>    |-  ( ph  ->  ( A. x  e.  X  E. c  e.  RR  A. t  e.  T  ( N `  ( t `
  x ) ) 
 <_  c  <->  E. d  e.  RR  A. t  e.  T  ( ( U normOp OLD W ) `  t )  <_  d ) )
 
Theoremubth 22223* Uniform Boundedness Theorem, also called the Banach-Steinhaus Theorem. Let  T be a collection of bounded linear operators on a Banach space. If, for every vector 
x, the norms of the operators' values are bounded, then the operators' norms are also bounded. Theorem 4.7-3 of [Kreyszig] p. 249. See also http://en.wikipedia.org/wiki/Uniform_boundedness_principle. (Contributed by NM, 7-Nov-2007.) (Proof shortened by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  N  =  (
 normCV `  W )   &    |-  M  =  ( U normOp OLD W )   =>    |-  ( ( U  e.  CBan  /\  W  e.  NrmCVec  /\  T  C_  ( U  BLnOp  W ) )  ->  ( A. x  e.  X  E. c  e.  RR  A. t  e.  T  ( N `  ( t `
  x ) ) 
 <_  c  <->  E. d  e.  RR  A. t  e.  T  ( M `  t ) 
 <_  d ) )
 
17.5.4  Minimizing Vector Theorem
 
Theoremminvecolem1 22224* Lemma for minveco 22234. The set of all distances from points of  Y to  A are a nonempty set of nonnegative reals. (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 ) ) )   =>    |-  ( ph  ->  ( R  C_  RR  /\  R  =/= 
 (/)  /\  A. w  e.  R  0  <_  w ) )
 
Theoremminvecolem2 22225* Lemma for minveco 22234. Any two points  K and 
L in  Y are close to each other if they are close to the infimum of distance to  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  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  B )   &    |-  ( ph  ->  K  e.  Y )   &    |-  ( ph  ->  L  e.  Y )   &    |-  ( ph  ->  ( ( A D K ) ^
 2 )  <_  (
 ( S ^ 2
 )  +  B ) )   &    |-  ( ph  ->  ( ( A D L ) ^ 2 )  <_  ( ( S ^
 2 )  +  B ) )   =>    |-  ( ph  ->  (
 ( K D L ) ^ 2 )  <_  ( 4  x.  B ) )
 
Theoremminvecolem3 22226* Lemma for minveco 22234. 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 22227* Lemma for minveco 22234. 
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 22228* Lemma for minveco 22234. 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 22229* Lemma for minveco 22234. 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 22230* Lemma for minveco 22234. 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 22231* Lemma for minveco 22234. Discharge the assumption about the sequence  F by applying countable choice ax-cc 8248. (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 22232* Lemma for minveco 22234. 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 22233* Lemma for minveco 22234. 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 22234* 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 ) ) )
 
17.6  Complex Hilbert spaces
 
17.6.1  Definition and basic properties
 
Syntaxchlo 22235 Extend class notation with the class of all complex Hilbert spaces.
 class  CHil OLD
 
Definitiondf-hlo 22236 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 22237 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 22238 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 22239 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 22240 The class of all complex Hilbert spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
 |- 
 Rel  CHil OLD
 
Theoremhlnv 22241 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 22242 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 22243 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 22244 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 22245 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 22246 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 22247 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
 ) ) ) )
 
17.6.2  Standard axioms for a complex Hilbert space
 
Theoremhlex 22248 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 22249 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 22250 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 22251 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 22252 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 22253 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 22254 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 22255 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 22256 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 22257 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 22258 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 22259 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 22260 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 22261 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 22262 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 22263 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 22264 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 22265 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 ) )
 
17.6.3  Examples of complex Hilbert spaces
 
Theoremcnchl 22266 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
 
17.6.4  Subspaces
 
Theoremssphl 22267 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 )
 
17.6.5  Hellinger-Toeplitz Theorem
 
Theoremhtthlem 22268* Lemma for htth 22269. 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 22205, so by the Uniform Boundedness theorem ubth 22223, 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 22269* 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 )
 
PART 18  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)

This part contains the definitions and theorems used by the Hilbert Space Explorer http://us.metamath.org/mpeuni/mmhil.html. Because it axiomatizes a single complex Hilbert space whose existence is assumed, its usefulness is limited. For example, it cannot work with real or quaternion Hilbert spaces and it cannot study relationships between two Hilbert spaces. More information can be found on the Hilbert Space Explorer page. Future development should work with general Hilbert spaces as defined by df-hil 16854.

 
18.1  Axiomatization of complex pre-Hilbert spaces
 
18.1.1  Basic Hilbert space definitions
 
Syntaxchil 22270 Extend class notation with Hilbert vector space.
 class  ~H
 
Syntaxcva 22271 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 8926.
 class  +h
 
Syntaxcsm 22272 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 22273 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 3766.
 class  .ih
 
Syntaxcno 22274 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 22275 Extend class notation with zero vector in Hilbert space.
 class  0h
 
Syntaxcmv 22276 Extend class notation with vector subtraction in Hilbert space.
 class  -h
 
Syntaxccau 22277 Extend class notation with set of Cauchy sequences in Hilbert space.
 class  Cauchy
 
Syntaxchli 22278 Extend class notation with convergence relation in Hilbert space.
 class  ~~>v
 
Syntaxcsh 22279 Extend class notation with set of subspaces of a Hilbert space.
 class  SH
 
Syntaxcch 22280 Extend class notation with set of closed subspaces of a Hilbert space.
 class  CH
 
Syntaxcort 22281 Extend class notation with orthogonal complement in  CH.
 class  _|_
 
Syntaxcph 22282 Extend class notation with subspace sum in  CH.
 class  +H
 
Syntaxcspn 22283 Extend class notation with subspace span in  CH.
 class  span
 
Syntaxchj 22284 Extend class notation with join in  CH.
 class  vH
 
Syntaxchsup 22285 Extend class notation with supremum of a collection in  CH.
 class  \/H
 
Syntaxc0h 22286 Extend class notation with zero of  CH.
 class  0H
 
Syntaxccm 22287 Extend class notation with the commutes relation on a Hilbert lattice.
 class  C_H
 
Syntaxcpjh 22288 Extend class notation with set of projections on a Hilbert space.
 class  proj  h
 
Syntaxchos 22289 Extend class notation with sum of Hilbert space operators.
 class  +op
 
Syntaxchot 22290 Extend class notation with scalar product of a Hilbert space operator.
 class  .op
 
Syntaxchod 22291 Extend class notation with difference of Hilbert space operators.
 class  -op
 
Syntaxchfs 22292 Extend class notation with sum of Hilbert space functionals.
 class  +fn
 
Syntaxchft 22293 Extend class notation with scalar product of Hilbert space functional.
 class  .fn
 
Syntaxch0o 22294 Extend class notation with the Hilbert space zero operator.
 class  0hop
 
Syntaxchio 22295 Extend class notation with Hilbert space identity operator.
 class  Iop
 
Syntaxcnop 22296 Extend class notation with the operator norm function.
 class  normop
 
Syntaxccop 22297 Extend class notation with set of continuous Hilbert space operators.
 class  ConOp
 
Syntaxclo 22298 Extend class notation with set of linear Hilbert space operators.
 class  LinOp
 
Syntaxcbo 22299 Extend class notation with set of bounded linear operators.
 class  BndLinOp
 
Syntaxcuo 22300 Extend class notation with set of unitary Hilbert space operators.
 class  UniOp
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