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Theorem List for Metamath Proof Explorer - 21701-21800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnorm-i 21701 Theorem 3.3(i) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( ( normh `  A )  =  0  <->  A  =  0h ) )
 
Theoremnormne0 21702 A norm is nonzero iff its argument is a nonzero vector. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( ( normh `  A )  =/=  0  <->  A  =/=  0h )
 )
 
Theoremnormcli 21703 Real closure of the norm of a vector. (Contributed by NM, 30-Sep-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   =>    |-  ( normh `  A )  e.  RR
 
Theoremnormsqi 21704 The square of a norm. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   =>    |-  ( ( normh `  A ) ^ 2 )  =  ( A  .ih  A )
 
Theoremnorm-i-i 21705 Theorem 3.3(i) of [Beran] p. 97. (Contributed by NM, 5-Sep-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   =>    |-  ( ( normh `  A )  =  0  <->  A  =  0h )
 
Theoremnormsq 21706 The square of a norm. (Contributed by NM, 12-May-2005.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( ( normh `  A ) ^ 2 )  =  ( A  .ih  A ) )
 
Theoremnormsub0i 21707 Two vectors are equal iff the norm of their difference is zero. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  (
 ( normh `  ( A  -h  B ) )  =  0  <->  A  =  B )
 
Theoremnormsub0 21708 Two vectors are equal iff the norm of their difference is zero. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( normh `  ( A  -h  B ) )  =  0  <->  A  =  B ) )
 
Theoremnorm-ii-i 21709 Triangle inequality for norms. Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( normh `  ( A  +h  B ) )  <_  ( ( normh `  A )  +  ( normh `  B ) )
 
Theoremnorm-ii 21710 Triangle inequality for norms. Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( normh `  ( A  +h  B ) )  <_  ( ( normh `  A )  +  ( normh `  B ) ) )
 
Theoremnorm-iii-i 21711 Theorem 3.3(iii) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
 |-  A  e.  CC   &    |-  B  e.  ~H   =>    |-  ( normh `  ( A  .h  B ) )  =  ( ( abs `  A )  x.  ( normh `  B ) )
 
Theoremnorm-iii 21712 Theorem 3.3(iii) of [Beran] p. 97. (Contributed by NM, 25-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  ~H )  ->  ( normh `  ( A  .h  B ) )  =  ( ( abs `  A )  x.  ( normh `  B ) ) )
 
Theoremnormsubi 21713 Negative doesn't change the norm of a Hilbert space vector. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( normh `  ( A  -h  B ) )  =  ( normh `  ( B  -h  A ) )
 
Theoremnormpythi 21714 Analogy to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  (
 ( A  .ih  B )  =  0  ->  ( ( normh `  ( A  +h  B ) ) ^
 2 )  =  ( ( ( normh `  A ) ^ 2 )  +  ( ( normh `  B ) ^ 2 ) ) )
 
Theoremnormsub 21715 Swapping order of subtraction doesn't change the norm of a vector. (Contributed by NM, 14-Aug-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( normh `  ( A  -h  B ) )  =  ( normh `  ( B  -h  A ) ) )
 
Theoremnormneg 21716 The norm of a vector equals the norm of its negative. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  (
 normh `  ( -u 1  .h  A ) )  =  ( normh `  A )
 )
 
Theoremnormpyth 21717 Analogy to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  .ih  B )  =  0  ->  ( ( normh `  ( A  +h  B ) ) ^ 2 )  =  ( ( ( normh `  A ) ^ 2
 )  +  ( (
 normh `  B ) ^
 2 ) ) ) )
 
Theoremnormpyc 21718 Corollary to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 26-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  .ih  B )  =  0  ->  ( normh `  A )  <_  ( normh `  ( A  +h  B ) ) ) )
 
Theoremnorm3difi 21719 Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  C  e.  ~H   =>    |-  ( normh `  ( A  -h  B ) )  <_  ( ( normh `  ( A  -h  C ) )  +  ( normh `  ( C  -h  B ) ) )
 
Theoremnorm3adifii 21720 Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 30-Sep-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  C  e.  ~H   =>    |-  ( abs `  (
 ( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) ) )  <_  ( normh `  ( A  -h  B ) )
 
Theoremnorm3lem 21721 Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  C  e.  ~H   &    |-  D  e.  RR   =>    |-  (
 ( ( normh `  ( A  -h  C ) )  <  ( D  / 
 2 )  /\  ( normh `  ( C  -h  B ) )  < 
 ( D  /  2
 ) )  ->  ( normh `  ( A  -h  B ) )  <  D )
 
Theoremnorm3dif 21722 Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 20-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( normh `  ( A  -h  B ) )  <_  ( ( normh `  ( A  -h  C ) )  +  ( normh `  ( C  -h  B ) ) ) )
 
Theoremnorm3dif2 21723 Norm of differences around common element. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( normh `  ( A  -h  B ) )  <_  ( ( normh `  ( C  -h  A ) )  +  ( normh `  ( C  -h  B ) ) ) )
 
Theoremnorm3lemt 21724 Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  ~H 
 /\  B  e.  ~H )  /\  ( C  e.  ~H 
 /\  D  e.  RR ) )  ->  ( ( ( normh `  ( A  -h  C ) )  < 
 ( D  /  2
 )  /\  ( normh `  ( C  -h  B ) )  <  ( D 
 /  2 ) ) 
 ->  ( normh `  ( A  -h  B ) )  <  D ) )
 
Theoremnorm3adifi 21725 Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 3-Oct-1999.) (New usage is discouraged.)
 |-  C  e.  ~H   =>    |-  ( ( A  e.  ~H 
 /\  B  e.  ~H )  ->  ( abs `  (
 ( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) ) )  <_  ( normh `  ( A  -h  B ) ) )
 
Theoremnormpari 21726 Parallelogram law for norms. Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  (
 ( ( normh `  ( A  -h  B ) ) ^ 2 )  +  ( ( normh `  ( A  +h  B ) ) ^ 2 ) )  =  ( ( 2  x.  ( ( normh `  A ) ^ 2
 ) )  +  (
 2  x.  ( (
 normh `  B ) ^
 2 ) ) )
 
Theoremnormpar 21727 Parallelogram law for norms. Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( ( normh `  ( A  -h  B ) ) ^ 2
 )  +  ( (
 normh `  ( A  +h  B ) ) ^
 2 ) )  =  ( ( 2  x.  ( ( normh `  A ) ^ 2 ) )  +  ( 2  x.  ( ( normh `  B ) ^ 2 ) ) ) )
 
Theoremnormpar2i 21728 Corollary of parallelogram law for norms. Part of Lemma 3.6 of [Beran] p. 100. (Contributed by NM, 5-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  C  e.  ~H   =>    |-  ( ( normh `  ( A  -h  B ) ) ^ 2 )  =  ( ( ( 2  x.  ( ( normh `  ( A  -h  C ) ) ^ 2
 ) )  +  (
 2  x.  ( (
 normh `  ( B  -h  C ) ) ^
 2 ) ) )  -  ( ( normh `  ( ( A  +h  B )  -h  (
 2  .h  C ) ) ) ^ 2
 ) )
 
Theorempolid2i 21729 Generalized polarization identity. Generalization of Exercise 4(a) of [ReedSimon] p. 63. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  C  e.  ~H   &    |-  D  e.  ~H   =>    |-  ( A  .ih  B )  =  ( ( ( ( ( A  +h  C )  .ih  ( D  +h  B ) )  -  ( ( A  -h  C )  .ih  ( D  -h  B ) ) )  +  ( _i 
 x.  ( ( ( A  +h  ( _i 
 .h  C ) ) 
 .ih  ( D  +h  ( _i  .h  B ) ) )  -  ( ( A  -h  ( _i  .h  C ) )  .ih  ( D  -h  ( _i  .h  B ) ) ) ) ) )  / 
 4 )
 
Theorempolidi 21730 Polarization identity. Recovers inner product from norm. Exercise 4(a) of [ReedSimon] p. 63. The outermost operation is + instead of - due to our mathematicians' (rather than physicists') version of axiom ax-his3 21656. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( A  .ih  B )  =  ( ( ( ( ( normh `  ( A  +h  B ) ) ^
 2 )  -  (
 ( normh `  ( A  -h  B ) ) ^
 2 ) )  +  ( _i  x.  (
 ( ( normh `  ( A  +h  ( _i  .h  B ) ) ) ^ 2 )  -  ( ( normh `  ( A  -h  ( _i  .h  B ) ) ) ^ 2 ) ) ) )  /  4
 )
 
Theorempolid 21731 Polarization identity. Recovers inner product from norm. Exercise 4(a) of [ReedSimon] p. 63. The outermost operation is + instead of - due to our mathematicians' (rather than physicists') version of axiom ax-his3 21656. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  B )  =  ( (
 ( ( ( normh `  ( A  +h  B ) ) ^ 2
 )  -  ( (
 normh `  ( A  -h  B ) ) ^
 2 ) )  +  ( _i  x.  (
 ( ( normh `  ( A  +h  ( _i  .h  B ) ) ) ^ 2 )  -  ( ( normh `  ( A  -h  ( _i  .h  B ) ) ) ^ 2 ) ) ) )  /  4
 ) )
 
17.2.3  Relate Hilbert space to normed complex vector spaces
 
Theoremhilablo 21732 Hilbert space vector addition is an Abelian group operation. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
 |-  +h  e.  AbelOp
 
Theoremhilid 21733 The group identity element of Hilbert space vector addition is the zero vector. (Contributed by NM, 16-Apr-2007.) (New usage is discouraged.)
 |-  (GId ` 
 +h  )  =  0h
 
Theoremhilvc 21734 Hilbert space is a complex vector space. Vector addition is  +h, and scalar product is  .h. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
 |-  <.  +h  ,  .h  >.  e.  CVec OLD
 
Theoremhilnormi 21735 Hilbert space norm in terms of vector space norm. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  ~H  =  ( BaseSet `  U )   &    |-  .ih  =  ( .i OLD `  U )   &    |-  U  e.  NrmCVec   =>    |- 
 normh  =  ( normCV `  U )
 
Theoremhilhhi 21736 Deduce the structure of Hilbert space from its components. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  ~H  =  ( BaseSet `  U )   &    |-  +h  =  ( +v `  U )   &    |- 
 .h  =  ( .s
 OLD `  U )   &    |-  .ih  =  ( .i OLD `  U )   &    |-  U  e.  NrmCVec   =>    |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.
 
Theoremhhnv 21737 Hilbert space is a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   =>    |-  U  e.  NrmCVec
 
Theoremhhva 21738 The group (addition) operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   =>    |- 
 +h  =  ( +v
 `  U )
 
Theoremhhba 21739 The base set of Hilbert space. This theorem provides an independent proof of df-hba 21542 (see comments in that definition). (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   =>    |- 
 ~H  =  ( BaseSet `  U )
 
Theoremhh0v 21740 The zero vector of Hilbert space. (Contributed by NM, 17-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   =>    |- 
 0h  =  ( 0vec `  U )
 
Theoremhhsm 21741 The scalar product operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   =>    |- 
 .h  =  ( .s
 OLD `  U )
 
Theoremhhvs 21742 The vector subtraction operation of Hilbert space. (Contributed by NM, 13-Dec-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   =>    |- 
 -h  =  ( -v
 `  U )
 
Theoremhhnm 21743 The norm function of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   =>    |- 
 normh  =  ( normCV `  U )
 
Theoremhhims 21744 The induced metric of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  D  =  ( normh  o. 
 -h  )   =>    |-  D  =  ( IndMet `  U )
 
Theoremhhims2 21745 Hilbert space distance metric. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  D  =  ( IndMet `  U )   =>    |-  D  =  ( normh  o. 
 -h  )
 
Theoremhhmet 21746 The induced metric of Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  D  =  ( IndMet `  U )   =>    |-  D  e.  ( Met `  ~H )
 
Theoremhhxmet 21747 The induced metric of Hilbert space. (Contributed by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  D  =  ( IndMet `  U )   =>    |-  D  e.  ( * Met `  ~H )
 
Theoremhhmetdval 21748 Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  D  =  ( IndMet `  U )   =>    |-  ( ( A  e.  ~H 
 /\  B  e.  ~H )  ->  ( A D B )  =  ( normh `  ( A  -h  B ) ) )
 
Theoremhhip 21749 The inner product operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   =>    |- 
 .ih  =  ( .i OLD `  U )
 
Theoremhhph 21750 The Hilbert space of the Hilbert Space Explorer is an inner product space. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   =>    |-  U  e.  CPreHil OLD
 
17.2.4  Bunjakovaskij-Cauchy-Schwarz inequality
 
TheorembcsiALT 21751 Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( abs `  ( A  .ih  B ) )  <_  (
 ( normh `  A )  x.  ( normh `  B )
 )
 
TheorembcsiHIL 21752 Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Proved from ZFC version.) (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( abs `  ( A  .ih  B ) )  <_  (
 ( normh `  A )  x.  ( normh `  B )
 )
 
Theorembcs 21753 Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( abs `  ( A  .ih  B ) ) 
 <_  ( ( normh `  A )  x.  ( normh `  B ) ) )
 
Theorembcs2 21754 Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 21752. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  ( normh `  A )  <_  1 )  ->  ( abs `  ( A  .ih  B ) )  <_  ( normh `  B ) )
 
Theorembcs3 21755 Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 21752. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  ( normh `  B )  <_  1 )  ->  ( abs `  ( A  .ih  B ) )  <_  ( normh `  A ) )
 
17.3  Cauchy sequences and completeness axiom
 
17.3.1  Cauchy sequences and limits
 
Theoremhcau 21756* Member of the set of Cauchy sequences on a Hilbert space. Definition for Cauchy sequence in [Beran] p. 96. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ( F  e.  Cauchy  <->  ( F : NN
 --> ~H  /\  A. x  e.  RR+  E. y  e. 
 NN  A. z  e.  ( ZZ>=
 `  y ) (
 normh `  ( ( F `
  y )  -h  ( F `  z ) ) )  <  x ) )
 
Theoremhcauseq 21757 A Cauchy sequences on a Hilbert space is a sequence. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ( F  e.  Cauchy  ->  F : NN --> ~H )
 
Theoremhcaucvg 21758* A Cauchy sequence on a Hilbert space converges. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  (
 ( F  e.  Cauchy  /\  A  e.  RR+ )  ->  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  y )  -h  ( F `  z ) ) )  <  A )
 
Theoremseq1hcau 21759* A sequence on a Hilbert space is a Cauchy sequence if it converges. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ( F : NN --> ~H  ->  ( F  e.  Cauchy  <->  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  y )  -h  ( F `  z ) ) )  <  x ) )
 
Theoremhlimi 21760* Express the predicate: The limit of vector sequence  F in a Hilbert space is  A, i.e.  F converges to  A. This means that for any real  x, no matter how small, there always exists an integer  y such that the norm of any later vector in the sequence minus the limit is less than  x. Definition of converge in [Beran] p. 96. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( F  ~~>v  A  <->  ( ( F : NN --> ~H  /\  A  e.  ~H )  /\  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  z )  -h  A ) )  <  x ) )
 
Theoremhlimseqi 21761 A sequence with a limit on a Hilbert space is a sequence. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( F  ~~>v  A  ->  F : NN --> ~H )
 
Theoremhlimveci 21762 Closure of the limit of a sequence on Hilbert space. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( F  ~~>v  A  ->  A  e.  ~H )
 
Theoremhlimconvi 21763* Convergence of a sequence on a Hilbert space. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( ( F  ~~>v  A 
 /\  B  e.  RR+ )  ->  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  z )  -h  A ) )  <  B )
 
Theoremhlim2 21764* The limit of a sequence on a Hilbert space. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  (
 ( F : NN --> ~H  /\  A  e.  ~H )  ->  ( F  ~~>v  A  <->  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
 ( normh `  ( ( F `  z )  -h  A ) )  < 
 x ) )
 
Theoremhlimadd 21765* Limit of the sum of two sequences in a Hilbert vector space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  F : NN --> ~H )   &    |-  ( ph  ->  G : NN --> ~H )   &    |-  ( ph  ->  F  ~~>v  A )   &    |-  ( ph  ->  G  ~~>v  B )   &    |-  H  =  ( n  e.  NN  |->  ( ( F `
  n )  +h  ( G `  n ) ) )   =>    |-  ( ph  ->  H  ~~>v  ( A  +h  B ) )
 
17.3.2  Derivation of the completeness axiom from ZF set theory
 
Theoremhilmet 21766 The Hilbert space norm determines a metric space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
 |-  D  =  ( normh  o.  -h  )   =>    |-  D  e.  ( Met `  ~H )
 
Theoremhilxmet 21767 The Hilbert space norm determines a metric space. (Contributed by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
 |-  D  =  ( normh  o.  -h  )   =>    |-  D  e.  ( * Met `  ~H )
 
Theoremhilmetdval 21768 Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
 |-  D  =  ( normh  o.  -h  )   =>    |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A D B )  =  ( normh `  ( A  -h  B ) ) )
 
Theoremhilims 21769 Hilbert space distance metric. (Contributed by NM, 13-Sep-2007.) (New usage is discouraged.)
 |-  ~H  =  ( BaseSet `  U )   &    |-  +h  =  ( +v `  U )   &    |- 
 .h  =  ( .s
 OLD `  U )   &    |-  .ih  =  ( .i OLD `  U )   &    |-  D  =  ( IndMet `  U )   &    |-  U  e.  NrmCVec   =>    |-  D  =  (
 normh  o.  -h  )
 
Theoremhhcau 21770 The Cauchy sequences of Hilbert space. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  D  =  ( IndMet `  U )   =>    |-  Cauchy  =  ( ( Cau `  D )  i^i  ( ~H  ^m  NN ) )
 
Theoremhhlm 21771 The limit sequences of Hilbert space. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  D  =  ( IndMet `  U )   &    |-  J  =  (
 MetOpen `  D )   =>    |-  ~~>v  =  ( ( ~~> t `  J )  |`  ( ~H  ^m  NN ) )
 
Theoremhhcmpl 21772* Lemma used for derivation of the completeness axiom ax-hcompl 21774 from ZFC Hilbert space theory. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  D  =  ( IndMet `  U )   &    |-  J  =  (
 MetOpen `  D )   &    |-  ( F  e.  ( Cau `  D )  ->  E. x  e.  ~H  F ( ~~> t `  J ) x )   =>    |-  ( F  e.  Cauchy  ->  E. x  e.  ~H  F  ~~>v  x )
 
Theoremhilcompl 21773* Lemma used for derivation of the completeness axiom ax-hcompl 21774 from ZFC Hilbert space theory. The first 5 hypotheses would be satisfied by the definitions described in ax-hilex 21572; the 6th would be satisfied by eqid 2285; the 7th by a given fixed Hilbert space; and the last by theorem hlcompl 21487. (Contributed by NM, 13-Sep-2007.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ~H  =  ( BaseSet `  U )   &    |-  +h  =  ( +v `  U )   &    |- 
 .h  =  ( .s
 OLD `  U )   &    |-  .ih  =  ( .i OLD `  U )   &    |-  D  =  ( IndMet `  U )   &    |-  J  =  (
 MetOpen `  D )   &    |-  U  e.  CHil OLD   &    |-  ( F  e.  ( Cau `  D )  ->  E. x  e.  ~H  F ( ~~> t `  J ) x )   =>    |-  ( F  e.  Cauchy  ->  E. x  e.  ~H  F  ~~>v  x )
 
17.3.3  Completeness postulate for a Hilbert space
 
Axiomax-hcompl 21774* Completeness of a Hilbert space. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)
 |-  ( F  e.  Cauchy  ->  E. x  e.  ~H  F  ~~>v  x )
 
17.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces
 
Theoremhhcms 21775 The Hilbert space induced metric determines a complete metric space. (Contributed by NM, 10-Apr-2008.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  D  =  ( IndMet `  U )   =>    |-  D  e.  ( CMet ` 
 ~H )
 
Theoremhhhl 21776 The Hilbert space structure is a complex Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   =>    |-  U  e.  CHil OLD
 
Theoremhilcms 21777 The Hilbert space norm determines a complete metric space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
 |-  D  =  ( normh  o.  -h  )   =>    |-  D  e.  ( CMet `  ~H )
 
Theoremhilhl 21778 The Hilbert space of the Hilbert Space Explorer is a complex Hilbert space. (Contributed by Steve Rodriguez, 29-Apr-2007.) (New usage is discouraged.)
 |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  CHil OLD
 
17.4  Subspaces and projections
 
17.4.1  Subspaces
 
Definitiondf-sh 21779 Define the set of subspaces of a Hilbert space. See issh 21780 for its membership relation. Basically, a subspace is a subset of a Hilbert space that acts like a vector space. From Definition of [Beran] p. 95. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  SH  =  { h  e.  ~P ~H  |  ( 0h  e.  h  /\  (  +h  " ( h  X.  h ) ) 
 C_  h  /\  (  .h  " ( CC  X.  h ) )  C_  h ) }
 
Theoremissh 21780 Subspace  H of a Hilbert space. A subspace is a subset of Hilbert space which contains the zero vector and is closed under vector addition and scalar multiplication. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  ( H  e.  SH  <->  ( ( H 
 C_  ~H  /\  0h  e.  H )  /\  ( (  +h  " ( H  X.  H ) ) 
 C_  H  /\  (  .h  " ( CC  X.  H ) )  C_  H ) ) )
 
Theoremissh2 21781* Subspace  H of a Hilbert space. A subspace is a subset of Hilbert space which contains the zero vector and is closed under vector addition and scalar multiplication. Definition of [Beran] p. 95. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  ( H  e.  SH  <->  ( ( H 
 C_  ~H  /\  0h  e.  H )  /\  ( A. x  e.  H  A. y  e.  H  ( x  +h  y )  e.  H  /\  A. x  e.  CC  A. y  e.  H  ( x  .h  y )  e.  H ) ) )
 
Theoremshss 21782 A subspace is a subset of Hilbert space. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  ( H  e.  SH  ->  H 
 C_  ~H )
 
Theoremshel 21783 A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( H  e.  SH  /\  A  e.  H ) 
 ->  A  e.  ~H )
 
Theoremshex 21784 The set of subspaces of a Hilbert space exists (is a set). (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
 |-  SH  e.  _V
 
Theoremshssii 21785 A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  SH   =>    |-  H  C_  ~H
 
Theoremsheli 21786 A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  SH   =>    |-  ( A  e.  H  ->  A  e.  ~H )
 
Theoremshelii 21787 A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  SH   &    |-  A  e.  H   =>    |-  A  e.  ~H
 
Theoremsh0 21788 The zero vector belongs to any subspace of a Hilbert space. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  ( H  e.  SH  ->  0h  e.  H )
 
Theoremshaddcl 21789 Closure of vector addition in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (New usage is discouraged.)
 |-  (
 ( H  e.  SH  /\  A  e.  H  /\  B  e.  H )  ->  ( A  +h  B )  e.  H )
 
Theoremshmulcl 21790 Closure of vector scalar multiplication in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (New usage is discouraged.)
 |-  (
 ( H  e.  SH  /\  A  e.  CC  /\  B  e.  H )  ->  ( A  .h  B )  e.  H )
 
TheoremshmulclOLD 21791 Closure of vector scalar multiplication in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( H  e.  SH  ->  ( ( A  e.  CC  /\  B  e.  H ) 
 ->  ( A  .h  B )  e.  H )
 )
 
Theoremissh3 21792* Subspace  H of a Hilbert space. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
 |-  ( H  C_  ~H  ->  ( H  e.  SH  <->  ( 0h  e.  H  /\  ( A. x  e.  H  A. y  e.  H  ( x  +h  y )  e.  H  /\  A. x  e.  CC  A. y  e.  H  ( x  .h  y )  e.  H ) ) ) )
 
Theoremshsubcl 21793 Closure of vector subtraction in a subspace of a Hilbert space. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( H  e.  SH  /\  A  e.  H  /\  B  e.  H )  ->  ( A  -h  B )  e.  H )
 
17.4.2  Closed subspaces
 
Definitiondf-ch 21794 Define the set of closed subspaces of a Hilbert space. A closed subspace is one in which the limit of every convergent sequence in the subspace belongs to the subspace. For its membership relation, see isch 21795. From Definition of [Beran] p. 107. Alternate definitions are given by isch2 21796 and isch3 21814. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)
 |-  CH  =  { h  e.  SH  |  (  ~~>v  " ( h  ^m  NN ) ) 
 C_  h }
 
Theoremisch 21795 Closed subspace  H of a Hilbert space. (Contributed by NM, 17-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  ( H  e.  CH  <->  ( H  e.  SH  /\  (  ~~>v  " ( H  ^m  NN ) ) 
 C_  H ) )
 
Theoremisch2 21796* Closed subspace  H of a Hilbert space. Definition of [Beran] p. 107. (Contributed by NM, 17-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  ( H  e.  CH  <->  ( H  e.  SH  /\  A. f A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )
 ) )
 
Theoremchsh 21797 A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  ( H  e.  CH  ->  H  e.  SH )
 
Theoremchsssh 21798 Closed subspaces are subspaces in a Hilbert space. (Contributed by NM, 29-May-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  CH  C_  SH
 
Theoremchex 21799 The set of closed subspaces of a Hilbert space exists (is a set). (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
 |-  CH  e.  _V
 
Theoremchshii 21800 A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  H  e.  SH
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