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Theorem List for Metamath Proof Explorer - 25901-26000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem0oval 25901 Value of the zero operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  W )   &    |-  O  =  ( U  0op  W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  A  e.  X )  ->  ( O `  A )  =  Z )
 
Theorem0oo 25902 The zero operator is an operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  Z  =  ( U  0op  W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  Z : X --> Y )
 
Theorem0lno 25903 The zero operator is linear. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
 |-  Z  =  ( U 
 0op  W )   &    |-  L  =  ( U  LnOp  W )   =>    |-  (
 ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  Z  e.  L )
 
Theoremnmoo0 25904 The operator norm of the zero operator. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.)
 |-  N  =  ( U
 normOpOLD W )   &    |-  Z  =  ( U  0op  W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( N `  Z )  =  0 )
 
Theorem0blo 25905 The zero operator is a bounded linear operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
 |-  Z  =  ( U 
 0op  W )   &    |-  B  =  ( U  BLnOp  W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  Z  e.  B )
 
Theoremnmlno0lem 25906 Lemma for nmlno0i 25907. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
 |-  N  =  ( U
 normOpOLD W )   &    |-  Z  =  ( U  0op  W )   &    |-  L  =  ( U 
 LnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e. 
 NrmCVec   &    |-  T  e.  L   &    |-  X  =  (
 BaseSet `  U )   &    |-  Y  =  ( BaseSet `  W )   &    |-  R  =  ( .sOLD `  U )   &    |-  S  =  ( .sOLD `  W )   &    |-  P  =  ( 0vec `  U )   &    |-  Q  =  ( 0vec `  W )   &    |-  K  =  ( normCV `  U )   &    |-  M  =  ( normCV `  W )   =>    |-  ( ( N `
  T )  =  0  <->  T  =  Z )
 
Theoremnmlno0i 25907 The norm of a linear operator is zero iff the operator is zero. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.)
 |-  N  =  ( U
 normOpOLD W )   &    |-  Z  =  ( U  0op  W )   &    |-  L  =  ( U 
 LnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e. 
 NrmCVec   =>    |-  ( T  e.  L  ->  ( ( N `  T )  =  0  <->  T  =  Z ) )
 
Theoremnmlno0 25908 The norm of a linear operator is zero iff the operator is zero. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
 |-  N  =  ( U
 normOpOLD W )   &    |-  Z  =  ( U  0op  W )   &    |-  L  =  ( U 
 LnOp  W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  (
 ( N `  T )  =  0  <->  T  =  Z ) )
 
Theoremnmlnoubi 25909* An upper bound for the operator norm of a linear operator, using only the properties of nonzero arguments. (Contributed by NM, 1-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   &    |-  K  =  ( normCV `  U )   &    |-  M  =  ( normCV `  W )   &    |-  N  =  ( U normOpOLD W )   &    |-  L  =  ( U 
 LnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e. 
 NrmCVec   =>    |-  ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A )  /\  A. x  e.  X  ( x  =/= 
 Z  ->  ( M `  ( T `  x ) )  <_  ( A  x.  ( K `  x ) ) ) )  ->  ( N `  T )  <_  A )
 
Theoremnmlnogt0 25910 The norm of a nonzero linear operator is positive. (Contributed by NM, 10-Dec-2007.) (New usage is discouraged.)
 |-  N  =  ( U
 normOpOLD W )   &    |-  Z  =  ( U  0op  W )   &    |-  L  =  ( U 
 LnOp  W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T  =/=  Z  <->  0  <  ( N `  T ) ) )
 
Theoremlnon0 25911* The domain of a nonzero linear operator contains a nonzero vector. (Contributed by NM, 15-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   &    |-  O  =  ( U  0op  W )   &    |-  L  =  ( U 
 LnOp  W )   =>    |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L ) 
 /\  T  =/=  O )  ->  E. x  e.  X  x  =/=  Z )
 
Theoremnmblolbii 25912 A lower bound for the norm of a bounded linear operator. (Contributed by NM, 7-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  L  =  (
 normCV `  U )   &    |-  M  =  (
 normCV `  W )   &    |-  N  =  ( U normOpOLD W )   &    |-  B  =  ( U  BLnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e.  NrmCVec   &    |-  T  e.  B   =>    |-  ( A  e.  X  ->  ( M `  ( T `  A ) )  <_  ( ( N `  T )  x.  ( L `  A ) ) )
 
Theoremnmblolbi 25913 A lower bound for the norm of a bounded linear operator. (Contributed by NM, 10-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  L  =  (
 normCV `  U )   &    |-  M  =  (
 normCV `  W )   &    |-  N  =  ( U normOpOLD W )   &    |-  B  =  ( U  BLnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e.  NrmCVec   =>    |-  ( ( T  e.  B  /\  A  e.  X )  ->  ( M `  ( T `  A ) )  <_  ( ( N `  T )  x.  ( L `  A ) ) )
 
Theoremisblo3i 25914* The predicate "is a bounded linear operator." Definition 2.7-1 of [Kreyszig] p. 91. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  (
 normCV `  U )   &    |-  N  =  (
 normCV `  W )   &    |-  L  =  ( U  LnOp  W )   &    |-  B  =  ( U  BLnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e.  NrmCVec   =>    |-  ( T  e.  B  <->  ( T  e.  L  /\  E. x  e.  RR  A. y  e.  X  ( N `  ( T `  y ) )  <_  ( x  x.  ( M `  y ) ) ) )
 
Theoremblo3i 25915* Properties that determine a bounded linear operator. (Contributed by NM, 13-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  (
 normCV `  U )   &    |-  N  =  (
 normCV `  W )   &    |-  L  =  ( U  LnOp  W )   &    |-  B  =  ( U  BLnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e.  NrmCVec   =>    |-  ( ( T  e.  L  /\  A  e.  RR  /\ 
 A. y  e.  X  ( N `  ( T `
  y ) ) 
 <_  ( A  x.  ( M `  y ) ) )  ->  T  e.  B )
 
Theoremblometi 25916 Upper bound for the distance between the values of a bounded linear operator. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  C  =  ( IndMet `  U )   &    |-  D  =  ( IndMet `  W )   &    |-  N  =  ( U normOpOLD W )   &    |-  B  =  ( U 
 BLnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e. 
 NrmCVec   =>    |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X ) 
 ->  ( ( T `  P ) D ( T `  Q ) )  <_  ( ( N `  T )  x.  ( P C Q ) ) )
 
Theoremblocnilem 25917 Lemma for blocni 25918 and lnocni 25919. If a linear operator is continuous at any point, it is bounded. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)
 |-  C  =  ( IndMet `  U )   &    |-  D  =  (
 IndMet `  W )   &    |-  J  =  ( MetOpen `  C )   &    |-  K  =  ( MetOpen `  D )   &    |-  L  =  ( U  LnOp  W )   &    |-  B  =  ( U  BLnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e. 
 NrmCVec   &    |-  T  e.  L   &    |-  X  =  (
 BaseSet `  U )   =>    |-  ( ( P  e.  X  /\  T  e.  ( ( J  CnP  K ) `  P ) )  ->  T  e.  B )
 
Theoremblocni 25918 A linear operator is continuous iff it is bounded. Theorem 2.7-9(a) of [Kreyszig] p. 97. (Contributed by NM, 18-Dec-2007.) (Revised by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)
 |-  C  =  ( IndMet `  U )   &    |-  D  =  (
 IndMet `  W )   &    |-  J  =  ( MetOpen `  C )   &    |-  K  =  ( MetOpen `  D )   &    |-  L  =  ( U  LnOp  W )   &    |-  B  =  ( U  BLnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e. 
 NrmCVec   &    |-  T  e.  L   =>    |-  ( T  e.  ( J  Cn  K )  <->  T  e.  B )
 
Theoremlnocni 25919 If a linear operator is continuous at any point, it is continuous everywhere. Theorem 2.7-9(b) of [Kreyszig] p. 97. (Contributed by NM, 18-Dec-2007.) (New usage is discouraged.)
 |-  C  =  ( IndMet `  U )   &    |-  D  =  (
 IndMet `  W )   &    |-  J  =  ( MetOpen `  C )   &    |-  K  =  ( MetOpen `  D )   &    |-  L  =  ( U  LnOp  W )   &    |-  B  =  ( U  BLnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e. 
 NrmCVec   &    |-  T  e.  L   &    |-  X  =  (
 BaseSet `  U )   =>    |-  ( ( P  e.  X  /\  T  e.  ( ( J  CnP  K ) `  P ) )  ->  T  e.  ( J  Cn  K ) )
 
Theoremblocn 25920 A linear operator is continuous iff it is bounded. Theorem 2.7-9(a) of [Kreyszig] p. 97. (Contributed by NM, 25-Dec-2007.) (New usage is discouraged.)
 |-  C  =  ( IndMet `  U )   &    |-  D  =  (
 IndMet `  W )   &    |-  J  =  ( MetOpen `  C )   &    |-  K  =  ( MetOpen `  D )   &    |-  B  =  ( U  BLnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e.  NrmCVec   &    |-  L  =  ( U 
 LnOp  W )   =>    |-  ( T  e.  L  ->  ( T  e.  ( J  Cn  K )  <->  T  e.  B ) )
 
Theoremblocn2 25921 A bounded linear operator is continuous. (Contributed by NM, 25-Dec-2007.) (New usage is discouraged.)
 |-  C  =  ( IndMet `  U )   &    |-  D  =  (
 IndMet `  W )   &    |-  J  =  ( MetOpen `  C )   &    |-  K  =  ( MetOpen `  D )   &    |-  B  =  ( U  BLnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e.  NrmCVec   =>    |-  ( T  e.  B  ->  T  e.  ( J  Cn  K ) )
 
Theoremajfval 25922* The adjoint function. (Contributed by NM, 25-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  P  =  ( .iOLD `  U )   &    |-  Q  =  ( .iOLD `  W )   &    |-  A  =  ( U adj W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  A  =  { <. t ,  s >.  |  (
 t : X --> Y  /\  s : Y --> X  /\  A. x  e.  X  A. y  e.  Y  (
 ( t `  x ) Q y )  =  ( x P ( s `  y ) ) ) } )
 
Theoremhmoval 25923* The set of Hermitian (self-adjoint) operators on a normed complex vector space. (Contributed by NM, 26-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  H  =  ( HmOp `  U )   &    |-  A  =  ( U adj U )   =>    |-  ( U  e.  NrmCVec  ->  H  =  { t  e.  dom  A  |  ( A `  t )  =  t } )
 
Theoremishmo 25924 The predicate "is a hermitian operator." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
 |-  H  =  ( HmOp `  U )   &    |-  A  =  ( U adj U )   =>    |-  ( U  e.  NrmCVec  ->  ( T  e.  H  <->  ( T  e.  dom 
 A  /\  ( A `  T )  =  T ) ) )
 
19.4  Inner product (pre-Hilbert) spaces
 
19.4.1  Definition and basic properties
 
Syntaxccphlo 25925 Extend class notation with the class of all complex inner product spaces (also called pre-Hilbert spaces).
 class  CPreHil OLD
 
Definitiondf-ph 25926* Define the class of all complex inner product spaces. An inner product space is a normed vector space whose norm satisfies the parallelogram law (a property that induces an inner product). Based on Exercise 4(b) of [ReedSimon] p. 63. The vector operation is  g, the scalar product is  s, and the norm is  n. An inner product space is also called a pre-Hilbert space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
 |-  CPreHil
 OLD  =  ( NrmCVec  i^i  { <. <. g ,  s >. ,  n >.  |  A. x  e.  ran  g A. y  e.  ran  g ( ( ( n `  ( x g y ) ) ^ 2 )  +  ( ( n `
  ( x g ( -u 1 s y ) ) ) ^
 2 ) )  =  ( 2  x.  (
 ( ( n `  x ) ^ 2
 )  +  ( ( n `  y ) ^ 2 ) ) ) } )
 
Theoremphnv 25927 Every complex inner product space is a normed complex vector space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
 |-  ( U  e.  CPreHil OLD 
 ->  U  e.  NrmCVec )
 
Theoremphrel 25928 The class of all complex inner product spaces is a relation. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
 |- 
 Rel  CPreHil OLD
 
Theoremphnvi 25929 Every complex inner product space is a normed complex vector space. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
 |-  U  e.  CPreHil OLD   =>    |-  U  e.  NrmCVec
 
Theoremisphg 25930* The predicate "is a complex inner product space." An inner product space is a normed vector space whose norm satisfies the parallelogram law. The vector (group) addition operation is  G, the scalar product is  S, and the norm is  N. An inner product space is also called a pre-Hilbert space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( G  e.  A  /\  S  e.  B  /\  N  e.  C )  ->  ( <. <. G ,  S >. ,  N >.  e.  CPreHil OLD  <->  (
 <. <. G ,  S >. ,  N >.  e.  NrmCVec  /\  A. x  e.  X  A. y  e.  X  (
 ( ( N `  ( x G y ) ) ^ 2 )  +  ( ( N `
  ( x G ( -u 1 S y ) ) ) ^
 2 ) )  =  ( 2  x.  (
 ( ( N `  x ) ^ 2
 )  +  ( ( N `  y ) ^ 2 ) ) ) ) ) )
 
Theoremphop 25931 A complex inner product space in terms of ordered pair components. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
 |-  G  =  ( +v
 `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( U  e.  CPreHil OLD 
 ->  U  =  <. <. G ,  S >. ,  N >. )
 
19.4.2  Examples of pre-Hilbert spaces
 
Theoremcncph 25932 The set of complex numbers is an inner product (pre-Hilbert) space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (Revised by Mario Carneiro, 7-Nov-2013.) (New usage is discouraged.)
 |-  U  =  <. <.  +  ,  x.  >. ,  abs >.   =>    |-  U  e.  CPreHil OLD
 
Theoremelimph 25933 Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   &    |-  U  e. 
 CPreHil OLD   =>    |- 
 if ( A  e.  X ,  A ,  Z )  e.  X
 
Theoremelimphu 25934 Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem. (Contributed by NM, 6-May-2007.) (New usage is discouraged.)
 |- 
 if ( U  e.  CPreHil OLD
 ,  U ,  <. <.  +  ,  x.  >. ,  abs >.
 )  e.  CPreHil OLD
 
19.4.3  Properties of pre-Hilbert spaces
 
Theoremisph 25935* The predicate "is an inner product space." (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  (
 normCV `  U )   =>    |-  ( U  e.  CPreHil OLD  <->  ( U  e.  NrmCVec  /\  A. x  e.  X  A. y  e.  X  ( ( ( N `  ( x G y ) ) ^ 2 )  +  ( ( N `  ( x M y ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `  x ) ^ 2 )  +  ( ( N `  y ) ^ 2
 ) ) ) ) )
 
Theoremphpar2 25936 The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  (
 normCV `  U )   =>    |-  ( ( U  e.  CPreHil 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 ) ) ) )
 
Theoremphpar 25937 The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  CPreHil 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
 ) ) ) )
 
Theoremip0i 25938 A slight variant of Equation 6.46 of [Ponnusamy] p. 362, where  J is either 1 or -1 to represent +-1. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  P  =  ( .iOLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   &    |-  C  e.  X   &    |-  N  =  ( normCV `  U )   &    |-  J  e.  CC   =>    |-  ( ( ( ( N `  ( ( A G B ) G ( J S C ) ) ) ^ 2 )  -  ( ( N `  ( ( A G B ) G (
 -u J S C ) ) ) ^
 2 ) )  +  ( ( ( N `
  ( ( A G ( -u 1 S B ) ) G ( J S C ) ) ) ^
 2 )  -  (
 ( N `  (
 ( A G (
 -u 1 S B ) ) G (
 -u J S C ) ) ) ^
 2 ) ) )  =  ( 2  x.  ( ( ( N `
  ( A G ( J S C ) ) ) ^ 2
 )  -  ( ( N `  ( A G ( -u J S C ) ) ) ^ 2 ) ) )
 
Theoremip1ilem 25939 Lemma for ip1i 25940. (Contributed by NM, 21-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  P  =  ( .iOLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   &    |-  C  e.  X   &    |-  N  =  ( normCV `  U )   &    |-  J  e.  CC   =>    |-  ( ( ( A G B ) P C )  +  (
 ( A G (
 -u 1 S B ) ) P C ) )  =  (
 2  x.  ( A P C ) )
 
Theoremip1i 25940 Equation 6.47 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  P  =  ( .iOLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   &    |-  C  e.  X   =>    |-  (
 ( ( A G B ) P C )  +  ( ( A G ( -u 1 S B ) ) P C ) )  =  ( 2  x.  ( A P C ) )
 
Theoremip2i 25941 Equation 6.48 of [Ponnusamy] p. 362. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  P  =  ( .iOLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   =>    |-  ( ( 2 S A ) P B )  =  ( 2  x.  ( A P B ) )
 
Theoremipdirilem 25942 Lemma for ipdiri 25943. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  P  =  ( .iOLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   &    |-  C  e.  X   =>    |-  (
 ( A G B ) P C )  =  ( ( A P C )  +  ( B P C ) )
 
Theoremipdiri 25943 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  P  =  ( .iOLD `  U )   &    |-  U  e.  CPreHil OLD   =>    |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) 
 ->  ( ( A G B ) P C )  =  ( ( A P C )  +  ( B P C ) ) )
 
Theoremipasslem1 25944 Lemma for ipassi 25954. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  P  =  ( .iOLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  B  e.  X   =>    |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) )
 
Theoremipasslem2 25945 Lemma for ipassi 25954. Show the inner product associative law for nonpositive integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  P  =  ( .iOLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  B  e.  X   =>    |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( (
 -u N S A ) P B )  =  ( -u N  x.  ( A P B ) ) )
 
Theoremipasslem3 25946 Lemma for ipassi 25954. Show the inner product associative law for all integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  P  =  ( .iOLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  B  e.  X   =>    |-  ( ( N  e.  ZZ  /\  A  e.  X )  ->  (
 ( N S A ) P B )  =  ( N  x.  ( A P B ) ) )
 
Theoremipasslem4 25947 Lemma for ipassi 25954. Show the inner product associative law for positive integer reciprocals. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  P  =  ( .iOLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  B  e.  X   =>    |-  ( ( N  e.  NN  /\  A  e.  X )  ->  (
 ( ( 1  /  N ) S A ) P B )  =  ( ( 1  /  N )  x.  ( A P B ) ) )
 
Theoremipasslem5 25948 Lemma for ipassi 25954. Show the inner product associative law for rational numbers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  P  =  ( .iOLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  B  e.  X   =>    |-  ( ( C  e.  QQ  /\  A  e.  X )  ->  (
 ( C S A ) P B )  =  ( C  x.  ( A P B ) ) )
 
Theoremipasslem7 25949* Lemma for ipassi 25954. Show that  ( ( w S A ) P B )  -  (
w  x.  ( A P B ) ) is continuous on  RR. (Contributed by NM, 23-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  P  =  ( .iOLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   &    |-  F  =  ( w  e.  RR  |->  ( ( ( w S A ) P B )  -  ( w  x.  ( A P B ) ) ) )   &    |-  J  =  ( topGen `  ran  (,) )   &    |-  K  =  ( TopOpen ` fld )   =>    |-  F  e.  ( J  Cn  K )
 
Theoremipasslem8 25950* Lemma for ipassi 25954. By ipasslem5 25948, 
F is 0 for all  QQ; since it is continuous and 
QQ is dense in  RR by qdensere2 21468, we conclude  F is 0 for all  RR. (Contributed by NM, 24-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  P  =  ( .iOLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   &    |-  F  =  ( w  e.  RR  |->  ( ( ( w S A ) P B )  -  ( w  x.  ( A P B ) ) ) )   =>    |-  F : RR --> { 0 }
 
Theoremipasslem9 25951 Lemma for ipassi 25954. Conclude from ipasslem8 25950 the inner product associative law for real numbers. (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  P  =  ( .iOLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   =>    |-  ( C  e.  RR  ->  ( ( C S A ) P B )  =  ( C  x.  ( A P B ) ) )
 
Theoremipasslem10 25952 Lemma for ipassi 25954. Show the inner product associative law for the imaginary number  _i. (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  P  =  ( .iOLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   &    |-  N  =  (
 normCV `  U )   =>    |-  ( ( _i S A ) P B )  =  ( _i  x.  ( A P B ) )
 
Theoremipasslem11 25953 Lemma for ipassi 25954. Show the inner product associative law for all complex numbers. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  P  =  ( .iOLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   =>    |-  ( C  e.  CC  ->  ( ( C S A ) P B )  =  ( C  x.  ( A P B ) ) )
 
Theoremipassi 25954 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  P  =  ( .iOLD `  U )   &    |-  U  e.  CPreHil OLD   =>    |-  ( ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) 
 ->  ( ( A S B ) P C )  =  ( A  x.  ( B P C ) ) )
 
Theoremdipdir 25955 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  P  =  ( .iOLD `  U )   =>    |-  ( ( U  e.  CPreHil OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( ( A G B ) P C )  =  ( ( A P C )  +  ( B P C ) ) )
 
Theoremdipdi 25956 Distributive law for inner product. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  P  =  ( .iOLD `  U )   =>    |-  ( ( U  e.  CPreHil OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( A P ( B G C ) )  =  ( ( A P B )  +  ( A P C ) ) )
 
Theoremip2dii 25957 Inner product of two sums. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  P  =  ( .iOLD `  U )   &    |-  U  e. 
 CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   &    |-  C  e.  X   &    |-  D  e.  X   =>    |-  ( ( A G B ) P ( C G D ) )  =  ( ( ( A P C )  +  ( B P D ) )  +  ( ( A P D )  +  ( B P C ) ) )
 
Theoremdipass 25958 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  P  =  ( .iOLD `  U )   =>    |-  ( ( U  e.  CPreHil OLD  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  ( ( A S B ) P C )  =  ( A  x.  ( B P C ) ) )
 
Theoremdipassr 25959 "Associative" law for second argument of inner product (compare dipass 25958). (Contributed by NM, 22-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  P  =  ( .iOLD `  U )   =>    |-  ( ( U  e.  CPreHil OLD  /\  ( A  e.  X  /\  B  e.  CC  /\  C  e.  X ) )  ->  ( A P ( B S C ) )  =  ( ( * `
  B )  x.  ( A P C ) ) )
 
Theoremdipassr2 25960 "Associative" law for inner product. Conjugate version of dipassr 25959. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  P  =  ( .iOLD `  U )   =>    |-  ( ( U  e.  CPreHil OLD  /\  ( A  e.  X  /\  B  e.  CC  /\  C  e.  X ) )  ->  ( A P ( ( * `  B ) S C ) )  =  ( B  x.  ( A P C ) ) )
 
Theoremdipsubdir 25961 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  P  =  ( .iOLD `  U )   =>    |-  ( ( U  e.  CPreHil OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( ( A M B ) P C )  =  ( ( A P C )  -  ( B P C ) ) )
 
Theoremdipsubdi 25962 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  P  =  ( .iOLD `  U )   =>    |-  ( ( U  e.  CPreHil OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( A P ( B M C ) )  =  ( ( A P B )  -  ( A P C ) ) )
 
Theorempythi 25963 The Pythagorean theorem for an arbitrary complex inner product (pre-Hilbert) space  U. The square of the norm of the sum of two orthogonal vectors (i.e. whose inner product is 0) is the sum of the squares of their norms. Problem 2 in [Kreyszig] p. 135. This is Metamath 100 proof #4. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  N  =  ( normCV `  U )   &    |-  P  =  ( .iOLD `  U )   &    |-  U  e. 
 CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   =>    |-  ( ( A P B )  =  0  ->  ( ( N `  ( A G B ) ) ^ 2 )  =  ( ( ( N `  A ) ^ 2 )  +  ( ( N `  B ) ^ 2
 ) ) )
 
Theoremsiilem1 25964 Lemma for sii 25967. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  N  =  (
 normCV `  U )   &    |-  P  =  ( .iOLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   &    |-  M  =  ( -v `  U )   &    |-  S  =  ( .sOLD `  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 ) ) )
 
Theoremsiilem2 25965 Lemma for sii 25967. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  N  =  (
 normCV `  U )   &    |-  P  =  ( .iOLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   &    |-  M  =  ( -v `  U )   &    |-  S  =  ( .sOLD `  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 25966 Inference from sii 25967. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  N  =  (
 normCV `  U )   &    |-  P  =  ( .iOLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   =>    |-  ( abs `  ( A P B ) ) 
 <_  ( ( N `  A )  x.  ( N `  B ) )
 
Theoremsii 25967 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 26235, bcsiALT 26294, bcsiHIL 26295, csbren 21992. This is Metamath 100 proof #78. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  N  =  (
 normCV `  U )   &    |-  P  =  ( .iOLD `  U )   &    |-  U  e.  CPreHil OLD   =>    |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( abs `  ( A P B ) ) 
 <_  ( ( N `  A )  x.  ( N `  B ) ) )
 
Theoremsspph 25968 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 25969* 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  =  ( .iOLD `  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 25970* 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  =  ( .iOLD `  U )   &    |-  U  e.  CPreHil OLD   =>    |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A. x  e.  X  ( x P A )  =  ( x P B )  <->  A  =  B )
 )
 
Theoremphoeqi 25971* A condition implying that two operators are equal. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .iOLD `  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 25972* Every operator has at most one adjoint. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .iOLD `  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 25973 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 25974 The adjoint function is a function. This is not immediately apparent from df-aj 25863 but results from the uniqueness shown by ajmoi 25972. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
 |-  A  =  ( U adj W )   =>    |-  ( ( U  e.  CPreHil OLD  /\  W  e.  NrmCVec )  ->  Fun  A )
 
Theoremajval 25975* Value of the adjoint function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  P  =  ( .iOLD `  U )   &    |-  Q  =  ( .iOLD `  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 ) ) ) ) )
 
19.5  Complex Banach spaces
 
19.5.1  Definition and basic properties
 
Syntaxccbn 25976 Extend class notation with the class of all complex Banach spaces.
 class  CBan
 
Definitiondf-cbn 25977 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 25978 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 25979 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 25980 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 25981 The class of all complex Banach spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
 |- 
 Rel  CBan
 
Theorembnsscmcl 25982 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 )
 ) )
 
19.5.2  Examples of complex Banach spaces
 
Theoremcnbn 25983 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
 
19.5.3  Uniform Boundedness Theorem
 
Theoremubthlem1 25984* Lemma for ubth 25987. 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 21935, 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 25985* Lemma for ubth 25987. 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
 normOpOLD W ) `  t )  <_  d )
 
Theoremubthlem3 25986* Lemma for ubth 25987. Prove the reverse implication, using nmblolbi 25913. (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 normOpOLD W ) `  t )  <_  d ) )
 
Theoremubth 25987* 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 normOpOLD 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 ) )
 
19.5.4  Minimizing Vector Theorem
 
Theoremminvecolem1 25988* Lemma for minveco 25998. 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 25989* Lemma for minveco 25998. 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 25990* Lemma for minveco 25998. 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 25991* Lemma for minveco 25998. 
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 25992* Lemma for minveco 25998. 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 25993* Lemma for minveco 25998. 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 25994* Lemma for minveco 25998. 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 25995* Lemma for minveco 25998. Discharge the assumption about the sequence  F by applying countable choice ax-cc 8806. (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 25996* Lemma for minveco 25998. 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 25997* Lemma for minveco 25998. 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 25998* 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 ) ) )
 
19.6  Complex Hilbert spaces
 
19.6.1  Definition and basic properties
 
Syntaxchlo 25999 Extend class notation with the class of all complex Hilbert spaces.
 class  CHilOLD
 
Definitiondf-hlo 26000 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.)
 |- 
 CHilOLD  =  ( CBan  i^i  CPreHil
 OLD )
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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 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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