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Theorem List for Metamath Proof Explorer - 26301-26400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremphrel 26301 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 26302 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 26303* 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 26304 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 26305 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 26306 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 26307 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 26308* 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 26309 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 26310 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 26311 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 26312 Lemma for ip1i 26313. (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 26313 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 26314 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 26315 Lemma for ipdiri 26316. (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 26316 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 26317 Lemma for ipassi 26327. 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 26318 Lemma for ipassi 26327. 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 26319 Lemma for ipassi 26327. 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 26320 Lemma for ipassi 26327. 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 26321 Lemma for ipassi 26327. 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 26322* Lemma for ipassi 26327. 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 26323* Lemma for ipassi 26327. By ipasslem5 26321, 
F is 0 for all  QQ; since it is continuous and 
QQ is dense in  RR by qdensere2 21726, 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 26324 Lemma for ipassi 26327. Conclude from ipasslem8 26323 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 26325 Lemma for ipassi 26327. 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 26326 Lemma for ipassi 26327. 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 26327 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 26328 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 26329 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 26330 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 26331 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 26332 "Associative" law for second argument of inner product (compare dipass 26331). (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 26333 "Associative" law for inner product. Conjugate version of dipassr 26332. (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 26334 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 26335 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 26336 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 26337 Lemma for sii 26340. (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 26338 Lemma for sii 26340. (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 26339 Inference from sii 26340. (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 26340 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 26608, bcsiALT 26667, bcsiHIL 26668, csbren 22246. 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 26341 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 26342* 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 26343* 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 26344* 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 26345* 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 26346 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 26347 The adjoint function is a function. This is not immediately apparent from df-aj 26236 but results from the uniqueness shown by ajmoi 26345. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
 |-  A  =  ( U adj W )   =>    |-  ( ( U  e.  CPreHil OLD  /\  W  e.  NrmCVec )  ->  Fun  A )
 
Theoremajval 26348* 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 26349 Extend class notation with the class of all complex Banach spaces.
 class  CBan
 
Definitiondf-cbn 26350 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 26351 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 26352 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 26353 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 26354 The class of all complex Banach spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
 |- 
 Rel  CBan
 
Theorembnsscmcl 26355 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 26356 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 26357* Lemma for ubth 26360. 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 22191, 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 26358* Lemma for ubth 26360. 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 26359* Lemma for ubth 26360. Prove the reverse implication, using nmblolbi 26286. (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 26360* 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 26361* Lemma for minveco 26371. 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 26362* Lemma for minveco 26371. 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 26363* Lemma for minveco 26371. 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 26364* Lemma for minveco 26371. 
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 26365* Lemma for minveco 26371. 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 26366* Lemma for minveco 26371. 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 26367* Lemma for minveco 26371. 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 26368* Lemma for minveco 26371. Discharge the assumption about the sequence  F by applying countable choice ax-cc 8863. (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 26369* Lemma for minveco 26371. 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 26370* Lemma for minveco 26371. 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 26371* 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 26372 Extend class notation with the class of all complex Hilbert spaces.
 class  CHilOLD
 
Definitiondf-hlo 26373 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 )
 
Theoremishlo 26374 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.  CHilOLD  <->  ( U  e.  CBan  /\  U  e. 
 CPreHil OLD ) )
 
Theoremhlobn 26375 Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
 |-  ( U  e.  CHilOLD 
 ->  U  e.  CBan )
 
Theoremhlph 26376 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.  CHilOLD 
 ->  U  e.  CPreHil OLD )
 
Theoremhlrel 26377 The class of all complex Hilbert spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
 |- 
 Rel  CHilOLD
 
Theoremhlnv 26378 Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
 |-  ( U  e.  CHilOLD 
 ->  U  e.  NrmCVec )
 
Theoremhlnvi 26379 Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
 |-  U  e.  CHilOLD   =>    |-  U  e.  NrmCVec
 
Theoremhlvc 26380 Every complex Hilbert space is a complex vector space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  W  =  ( 1st `  U )   =>    |-  ( U  e.  CHilOLD 
 ->  W  e.  CVecOLD )
 
Theoremhlcmet 26381 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.  CHilOLD  ->  D  e.  ( CMet `  X ) )
 
Theoremhlmet 26382 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.  CHilOLD  ->  D  e.  ( Met `  X ) )
 
Theoremhlpar2 26383 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.  CHilOLD  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( ( ( N `
  ( A G B ) ) ^
 2 )  +  (
 ( N `  ( A M B ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `
  A ) ^
 2 )  +  (
 ( N `  B ) ^ 2 ) ) ) )
 
Theoremhlpar 26384 The parallelogram law satified by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  CHilOLD  /\  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
 ) ) ) )
 
19.6.2  Standard axioms for a complex Hilbert space
 
Theoremhlex 26385 The base set of a Hilbert space is a set. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   =>    |-  X  e.  _V
 
Theoremhladdf 26386 Mapping for Hilbert space vector addition. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( U  e.  CHilOLD  ->  G : ( X  X.  X ) --> X )
 
Theoremhlcom 26387 Hilbert space vector addition is commutative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( ( U  e.  CHilOLD  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( A G B )  =  ( B G A ) )
 
Theoremhlass 26388 Hilbert space vector addition is associative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( ( U  e.  CHilOLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( ( A G B ) G C )  =  ( A G ( B G C ) ) )
 
Theoremhl0cl 26389 The Hilbert space zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   =>    |-  ( U  e.  CHilOLD  ->  Z  e.  X )
 
Theoremhladdid 26390 Hilbert space addition with the zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  ( ( U  e.  CHilOLD  /\  A  e.  X )  ->  ( A G Z )  =  A )
 
Theoremhlmulf 26391 Mapping for Hilbert space scalar multiplication. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .sOLD `  U )   =>    |-  ( U  e.  CHilOLD 
 ->  S : ( CC 
 X.  X ) --> X )
 
Theoremhlmulid 26392 Hilbert space scalar multiplication by one. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .sOLD `  U )   =>    |-  ( ( U  e.  CHilOLD  /\  A  e.  X )  ->  ( 1 S A )  =  A )
 
Theoremhlmulass 26393 Hilbert space scalar multiplication associative law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .sOLD `  U )   =>    |-  ( ( U  e.  CHilOLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X ) )  ->  ( ( A  x.  B ) S C )  =  ( A S ( B S C ) ) )
 
Theoremhldi 26394 Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .sOLD `  U )   =>    |-  ( ( U  e.  CHilOLD  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  ( A S ( B G C ) )  =  ( ( A S B ) G ( A S C ) ) )
 
Theoremhldir 26395 Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .sOLD `  U )   =>    |-  ( ( U  e.  CHilOLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X ) )  ->  ( ( A  +  B ) S C )  =  ( ( A S C ) G ( B S C ) ) )
 
Theoremhlmul0 26396 Hilbert space scalar multiplication by zero. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  ( ( U  e.  CHilOLD  /\  A  e.  X )  ->  ( 0 S A )  =  Z )
 
Theoremhlipf 26397 Mapping for Hilbert space inner product. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .iOLD `  U )   =>    |-  ( U  e.  CHilOLD 
 ->  P : ( X  X.  X ) --> CC )
 
Theoremhlipcj 26398 Conjugate law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .iOLD `  U )   =>    |-  ( ( U  e.  CHilOLD  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( A P B )  =  ( * `  ( B P A ) ) )
 
Theoremhlipdir 26399 Distributive law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  P  =  ( .iOLD `  U )   =>    |-  ( ( U  e.  CHilOLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( ( A G B ) P C )  =  ( ( A P C )  +  ( B P C ) ) )
 
Theoremhlipass 26400 Associative law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .sOLD `  U )   &    |-  P  =  ( .iOLD `  U )   =>    |-  ( ( U  e.  CHilOLD  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  ( ( A S B ) P C )  =  ( A  x.  ( B P C ) ) )
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