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Theorem List for Metamath Proof Explorer - 21401-21500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremphpar2 21401 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 21402 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 21403 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 21404 Lemma for ip1i 21405. (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 21405 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 21406 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 21407 Lemma for ipdiri 21408. (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 21408 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 21409 Lemma for ipassi 21419. 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 21410 Lemma for ipassi 21419. 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 21411 Lemma for ipassi 21419. 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 21412 Lemma for ipassi 21419. 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 21413 Lemma for ipassi 21419. 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 21414* Lemma for ipassi 21419. 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 21415* Lemma for ipassi 21419. By ipasslem5 21413, 
F is 0 for all  QQ; since it is continuous and 
QQ is dense in  RR by qdensere2 18303, 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 21416 Lemma for ipassi 21419. Conclude from ipasslem8 21415 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 21417 Lemma for ipassi 21419. 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 21418 Lemma for ipassi 21419. 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 21419 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 21420 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 21421 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 21422 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 21423 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 21424 "Associative" law for second argument of inner product (compare dipass 21423). (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 21425 "Associative" law for inner product. Conjugate version of dipassr 21424. (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 21426 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 21427 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 21428 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 21429 Lemma for sii 21432. (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 21430 Lemma for sii 21432. (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 21431 Inference from sii 21432. (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 21432 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 21699, bcsiALT 21758, bcsiHIL 21759, csbrn 26462. (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 21433 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 21434* 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 21435* 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 21436* 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 21437* 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 21438 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 21439 The adjoint function is a function. This is not immediately apparent from df-aj 21328 but results from the uniqueness shown by ajmoi 21437. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
 |-  A  =  ( U adj W )   =>    |-  ( ( U  e.  CPreHil OLD  /\  W  e.  NrmCVec )  ->  Fun  A )
 
Theoremajval 21440* 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 ) ) ) ) )
 
16.5  Complex Banach spaces
 
16.5.1  Definition and basic properties
 
Syntaxccbn 21441 Extend class notation with the class of all complex Banach spaces.
 class  CBan
 
Definitiondf-cbn 21442 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 21443 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 21444 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 21445 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 21446 The class of all complex Banach spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
 |- 
 Rel  CBan
 
Theorembnsscmcl 21447 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 )
 ) )
 
16.5.2  Examples of complex Banach spaces
 
Theoremcnbn 21448 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
 
16.5.3  Uniform Boundedness Theorem
 
Theoremubthlem1 21449* Lemma for ubth 21452. 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 18752, 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 21450* Lemma for ubth 21452. 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 21451* Lemma for ubth 21452. Prove the reverse implication, using nmblolbi 21378. (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 21452* 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 ) )
 
16.5.4  Minimizing Vector Theorem
 
Theoremminvecolem1 21453* Lemma for minveco 21463. 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 21454* Lemma for minveco 21463. 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 21455* Lemma for minveco 21463. 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 21456* Lemma for minveco 21463. 
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 21457* Lemma for minveco 21463. 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 21458* Lemma for minveco 21463. 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 21459* Lemma for minveco 21463. 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 21460* Lemma for minveco 21463. Discharge the assumption about the sequence  F by applying countable choice ax-cc 8061. (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 21461* Lemma for minveco 21463. 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 21462* Lemma for minveco 21463. 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 21463* 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 ) ) )
 
16.6  Complex Hilbert spaces
 
16.6.1  Definition and basic properties
 
Syntaxchlo 21464 Extend class notation with the class of all complex Hilbert spaces.
 class  CHil OLD
 
Definitiondf-hlo 21465 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 21466 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 21467 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 21468 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 21469 The class of all complex Hilbert spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
 |- 
 Rel  CHil OLD
 
Theoremhlnv 21470 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 21471 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 21472 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 21473 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 21474 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 21475 The parallelogram law satified by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  (
 normCV `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( ( ( N `
  ( A G B ) ) ^
 2 )  +  (
 ( N `  ( A M B ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `
  A ) ^
 2 )  +  (
 ( N `  B ) ^ 2 ) ) ) )
 
Theoremhlpar 21476 The parallelogram law satified by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  A  e.  X  /\  B  e.  X )  ->  ( ( ( N `  ( A G B ) ) ^ 2 )  +  ( ( N `  ( A G ( -u 1 S B ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `  A ) ^ 2 )  +  ( ( N `  B ) ^ 2
 ) ) ) )
 
16.6.2  Standard axioms for a complex Hilbert space
 
Theoremhlex 21477 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 21478 Mapping for Hilbert space vector addition. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( U  e.  CHil OLD  ->  G : ( X  X.  X ) --> X )
 
Theoremhlcom 21479 Hilbert space vector addition is commutative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( A G B )  =  ( B G A ) )
 
Theoremhlass 21480 Hilbert space vector addition is associative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( ( A G B ) G C )  =  ( A G ( B G C ) ) )
 
Theoremhl0cl 21481 The Hilbert space zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   =>    |-  ( U  e.  CHil OLD  ->  Z  e.  X )
 
Theoremhladdid 21482 Hilbert space addition with the zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  A  e.  X )  ->  ( A G Z )  =  A )
 
Theoremhlmulf 21483 Mapping for Hilbert space scalar multiplication. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( U  e.  CHil OLD 
 ->  S : ( CC 
 X.  X ) --> X )
 
Theoremhlmulid 21484 Hilbert space scalar multiplication by one. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  A  e.  X )  ->  ( 1 S A )  =  A )
 
Theoremhlmulass 21485 Hilbert space scalar multiplication associative law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X ) )  ->  ( ( A  x.  B ) S C )  =  ( A S ( B S C ) ) )
 
Theoremhldi 21486 Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  ( A S ( B G C ) )  =  ( ( A S B ) G ( A S C ) ) )
 
Theoremhldir 21487 Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X ) )  ->  ( ( A  +  B ) S C )  =  ( ( A S C ) G ( B S C ) ) )
 
Theoremhlmul0 21488 Hilbert space scalar multiplication by zero. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  A  e.  X )  ->  ( 0 S A )  =  Z )
 
Theoremhlipf 21489 Mapping for Hilbert space inner product. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( U  e.  CHil OLD 
 ->  P : ( X  X.  X ) --> CC )
 
Theoremhlipcj 21490 Conjugate law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( A P B )  =  ( * `  ( B P A ) ) )
 
Theoremhlipdir 21491 Distributive law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( ( A G B ) P C )  =  ( ( A P C )  +  ( B P C ) ) )
 
Theoremhlipass 21492 Associative law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i
 OLD `  U )   =>    |-  (
 ( U  e.  CHil OLD  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A S B ) P C )  =  ( A  x.  ( B P C ) ) )
 
Theoremhlipgt0 21493 The inner product of a Hilbert space vector by itself is positive. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  A  e.  X  /\  A  =/=  Z ) 
 ->  0  <  ( A P A ) )
 
Theoremhlcompl 21494 Completeness of a Hilbert space. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
 |-  D  =  ( IndMet `  U )   &    |-  J  =  (
 MetOpen `  D )   =>    |-  ( ( U  e.  CHil OLD  /\  F  e.  ( Cau `  D )
 )  ->  F  e.  dom  ( ~~> t `  J ) )
 
16.6.3  Examples of complex Hilbert spaces
 
Theoremcnchl 21495 The set of complex numbers is a complex Hilbert space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +  ,  x.  >. ,  abs >.   =>    |-  U  e.  CHil OLD
 
16.6.4  Subspaces
 
Theoremssphl 21496 A Banach subspace of an inner product space is a Hilbert space. (Contributed by NM, 11-Apr-2008.) (New usage is discouraged.)
 |-  H  =  ( SubSp `  U )   =>    |-  ( ( U  e.  CPreHil OLD  /\  W  e.  H  /\  W  e.  CBan )  ->  W  e.  CHil OLD )
 
16.6.5  Hellinger-Toeplitz Theorem
 
Theoremhtthlem 21497* Lemma for htth 21498. The collection  K, which consists of functions  F ( z ) ( w )  =  <. w  |  T
( z ) >.  =  <. T ( w )  |  z >. for each  z in the unit ball, is a collection of bounded linear functions by ipblnfi 21434, so by the Uniform Boundedness theorem ubth 21452, there is a uniform bound  y on  ||  F ( x )  || for all  x in the unit ball. Then  |  T (
x )  |  ^
2  =  <. T ( x )  |  T
( x ) >.  =  F ( x ) (  T ( x ) )  <_  y  |  T ( x )  |, so  |  T ( x )  |  <_  y and 
T is bounded. (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  L  =  ( U 
 LnOp  U )   &    |-  B  =  ( U  BLnOp  U )   &    |-  N  =  ( normCV `  U )   &    |-  U  e.  CHil OLD   &    |-  W  =  <. <.  +  ,  x.  >. ,  abs >.   &    |-  ( ph  ->  T  e.  L )   &    |-  ( ph  ->  A. x  e.  X  A. y  e.  X  ( x P ( T `  y
 ) )  =  ( ( T `  x ) P y ) )   &    |-  F  =  ( z  e.  X  |->  ( w  e.  X  |->  ( w P ( T `  z
 ) ) ) )   &    |-  K  =  ( F " { z  e.  X  |  ( N `  z
 )  <_  1 }
 )   =>    |-  ( ph  ->  T  e.  B )
 
Theoremhtth 21498* Hellinger-Toeplitz Theorem: any self-adjoint linear operator defined on all of Hilbert space is bounded. Theorem 10.1-1 of [Kreyszig] p. 525. Discovered by E. Hellinger and O. Toeplitz in 1910, "it aroused both admiration and puzzlement since the theorem establishes a relation between properties of two different kinds, namely, the properties of being defined everywhere and being bounded." (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  L  =  ( U 
 LnOp  U )   &    |-  B  =  ( U  BLnOp  U )   =>    |-  ( ( U  e.  CHil OLD  /\  T  e.  L  /\  A. x  e.  X  A. y  e.  X  ( x P ( T `  y
 ) )  =  ( ( T `  x ) P y ) ) 
 ->  T  e.  B )
 
PART 17  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)

This part contains the definitions and theorems used by the Hilbert Space Explorer http://us.metamath.org/mpeuni/mmhil.html. Because it axiomatizes a single complex Hilbert space whose existence is assumed, its usefulness is limited. For example, it cannot work with real or quaternion Hilbert spaces and it cannot study relationships between two Hilbert spaces. More information can be found on the Hilbert Space Explorer page. Future development should work with general Hilbert spaces as defined by df-hil 16604.

 
17.1  Axiomatization of complex pre-Hilbert spaces
 
17.1.1  Basic Hilbert space definitions
 
Syntaxchil 21499 Extend class notation with Hilbert vector space.
 class  ~H
 
Syntaxcva 21500 Extend class notation with vector addition in Hilbert space. In the literature, the subscript "v" is omitted, but we need it to avoid ambiguity with complex number addition  + caddc 8740.
 class  +h
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