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Theorem List for Metamath Proof Explorer - 22301-22400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlnon0 22301* 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 22302 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 normOp OLD 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 22303 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 normOp OLD 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 22304* 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 22305* 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 22306 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 normOp OLD 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 22307 Lemma for blocni 22308 and lnocni 22309. 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 22308 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 22309 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 22310 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 22311 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 22312* 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  =  ( .i OLD `  U )   &    |-  Q  =  ( .i
 OLD `  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 22313* 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 22314 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 ) ) )
 
17.4  Inner product (pre-Hilbert) spaces
 
17.4.1  Definition and basic properties
 
Syntaxccphlo 22315 Extend class notation with the class of all complex inner product spaces (also called pre-Hilbert spaces).
 class  CPreHil OLD
 
Definitiondf-ph 22316* 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 22317 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 22318 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 22319 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 22320* 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 22321 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  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( U  e.  CPreHil OLD 
 ->  U  =  <. <. G ,  S >. ,  N >. )
 
17.4.2  Examples of pre-Hilbert spaces
 
Theoremcncph 22322 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 22323 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 22324 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
 
17.4.3  Properties of pre-Hilbert spaces
 
Theoremisph 22325* 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 22326 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 22327 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  =  ( .s OLD `  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 22328 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  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  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 22329 Lemma for ip1i 22330. (Contributed by NM, 21-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  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 22330 Equation 6.47 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  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 22331 Equation 6.48 of [Ponnusamy] p. 362. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   =>    |-  ( ( 2 S A ) P B )  =  ( 2  x.  ( A P B ) )
 
Theoremipdirilem 22332 Lemma for ipdiri 22333. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  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 22333 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  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  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 22334 Lemma for ipassi 22344. 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  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  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 22335 Lemma for ipassi 22344. 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  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  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 22336 Lemma for ipassi 22344. 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  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  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 22337 Lemma for ipassi 22344. 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  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  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 22338 Lemma for ipassi 22344. 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  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  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 22339* Lemma for ipassi 22344. 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  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  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 22340* Lemma for ipassi 22344. By ipasslem5 22338, 
F is 0 for all  QQ; since it is continuous and 
QQ is dense in  RR by qdensere2 18830, 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  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  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 22341 Lemma for ipassi 22344. Conclude from ipasslem8 22340 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  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  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 22342 Lemma for ipassi 22344. 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  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  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 22343 Lemma for ipassi 22344. 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  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  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 22344 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  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  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 22345 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  =  ( .i OLD `  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 22346 Distributive law for inner product. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  P  =  ( .i OLD `  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 22347 Inner product of two sums. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  P  =  ( .i OLD `  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 22348 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  =  ( .s OLD `  U )   &    |-  P  =  ( .i
 OLD `  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 22349 "Associative" law for second argument of inner product (compare dipass 22348). (Contributed by NM, 22-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i
 OLD `  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 22350 "Associative" law for inner product. Conjugate version of dipassr 22349. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i
 OLD `  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 22351 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  P  =  ( .i OLD `  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 22352 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  P  =  ( .i OLD `  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 22353 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. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  N  =  ( normCV `  U )   &    |-  P  =  ( .i
 OLD `  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 22354 Lemma for sii 22357. (Contributed by NM, 23-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 ) ) )
 
Theoremsiilem2 22355 Lemma for sii 22357. (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 22356 Inference from sii 22357. (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 22357 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 22624, bcsiALT 22683, bcsiHIL 22684, csbrn 26458. (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 22358 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 22359* 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 22360* 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 22361* 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 22362* 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 22363 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 22364 The adjoint function is a function. This is not immediately apparent from df-aj 22253 but results from the uniqueness shown by ajmoi 22362. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
 |-  A  =  ( U adj W )   =>    |-  ( ( U  e.  CPreHil OLD  /\  W  e.  NrmCVec )  ->  Fun  A )
 
Theoremajval 22365* 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 22366 Extend class notation with the class of all complex Banach spaces.
 class  CBan
 
Definitiondf-cbn 22367 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 22368 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 22369 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 22370 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 22371 The class of all complex Banach spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
 |- 
 Rel  CBan
 
Theorembnsscmcl 22372 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 22373 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 22374* Lemma for ubth 22377. 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 19285, 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 22375* Lemma for ubth 22377. 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 22376* Lemma for ubth 22377. Prove the reverse implication, using nmblolbi 22303. (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 22377* 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 22378* Lemma for minveco 22388. 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 22379* Lemma for minveco 22388. 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 22380* Lemma for minveco 22388. 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 22381* Lemma for minveco 22388. 
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 22382* Lemma for minveco 22388. 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 22383* Lemma for minveco 22388. 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 22384* Lemma for minveco 22388. 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 22385* Lemma for minveco 22388. Discharge the assumption about the sequence  F by applying countable choice ax-cc 8317. (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 22386* Lemma for minveco 22388. 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 22387* Lemma for minveco 22388. 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 22388* 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 22389 Extend class notation with the class of all complex Hilbert spaces.
 class  CHil OLD
 
Definitiondf-hlo 22390 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 22391 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 22392 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 22393 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 22394 The class of all complex Hilbert spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
 |- 
 Rel  CHil OLD
 
Theoremhlnv 22395 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 22396 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 22397 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 22398 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 22399 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 22400 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 ) ) ) )
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