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Theorem List for Metamath Proof Explorer - 21701-21800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhhmetdval 21701 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 21702 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 21703 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
 
15.9.10  Bunjakovaskij-Cauchy-Schwarz inequality
 
TheorembcsiALT 21704 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 21705 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 21706 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 21707 Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 21705. (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 21708 Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 21705. (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 ) )
 
15.9.11  Cauchy sequences and limits
 
Theoremhcau 21709* 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 21710 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 21711* 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 21712* 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 21713* 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 21714 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 21715 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 21716* 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 21717* 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 21718* 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 ) )
 
15.9.12  Derivation of the completeness axiom from ZF set theory
 
Theoremhilmet 21719 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 21720 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 21721 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 21722 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 21723 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 21724 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 21725* Lemma used for derivation of the completeness axiom ax-hcompl 21727 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 21726* Lemma used for derivation of the completeness axiom ax-hcompl 21727 from ZFC Hilbert space theory. The first 5 hypotheses would be satisfied by the definitions described in ax-hilex 21525; the 6th would be satisfied by eqid 2256; the 7th by a given fixed Hilbert space; and the last by theorem hlcompl 21440. (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 )
 
15.9.13  Completeness postulate for a Hilbert space
 
Axiomax-hcompl 21727* 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 )
 
15.9.14  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces
 
Theoremhhcms 21728 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 21729 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 21730 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 21731 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
 
15.9.15  Subspaces
 
Definitiondf-sh 21732 Define the set of subspaces of a Hilbert space. See issh 21733 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 21733 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 21734* 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 21735 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 21736 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 21737 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 21738 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 21739 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 21740 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 21741 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 21742 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 21743 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 21744 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 21745* 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 21746 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 )
 
15.9.16  Closed subspaces
 
Definitiondf-ch 21747 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 21748. From Definition of [Beran] p. 107. Alternate definitions are given by isch2 21749 and isch3 21767. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)
 |-  CH  =  { h  e.  SH  |  (  ~~>v  " ( h  ^m  NN ) ) 
 C_  h }
 
Theoremisch 21748 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 21749* 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 21750 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 21751 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 21752 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 21753 A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  H  e.  SH
 
Theoremch0 21754 The zero vector belongs to any closed subspace of a Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.)
 |-  ( H  e.  CH  ->  0h  e.  H )
 
Theoremchss 21755 A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.)
 |-  ( H  e.  CH  ->  H  C_ 
 ~H )
 
Theoremchel 21756 A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  H ) 
 ->  A  e.  ~H )
 
Theoremchssii 21757 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.  CH   =>    |-  H  C_  ~H
 
Theoremcheli 21758 A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( A  e.  H  ->  A  e.  ~H )
 
Theoremchelii 21759 A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  H   =>    |-  A  e.  ~H
 
Theoremchlimi 21760 The limit property of a closed subspace of a Hilbert space. (Contributed by NM, 14-Sep-1999.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( ( H  e.  CH 
 /\  F : NN --> H  /\  F  ~~>v  A ) 
 ->  A  e.  H )
 
Theoremhlim0 21761 The zero sequence in Hilbert space converges to the zero vector. (Contributed by NM, 17-Aug-1999.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ( NN  X.  { 0h }
 )  ~~>v  0h
 
Theoremhlimcaui 21762 If a sequence in Hilbert space subset converges to a limit, it is a Cauchy sequence. (Contributed by NM, 17-Aug-1999.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ( F  ~~>v  A  ->  F  e.  Cauchy )
 
Theoremhlimf 21763 Function-like behavior of the convergence relation. (Contributed by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ~~>v  : dom  ~~>v  --> ~H
 
Theoremhlimuni 21764 A Hilbert space sequence converges to at most one limit. (Contributed by NM, 19-Aug-1999.) (Revised by Mario Carneiro, 2-May-2015.) (New usage is discouraged.)
 |-  (
 ( F  ~~>v  A  /\  F  ~~>v  B )  ->  A  =  B )
 
Theoremhlimreui 21765* The limit of a Hilbert space sequence is unique. (Contributed by NM, 19-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ( E. x  e.  H  F  ~~>v  x  <->  E! x  e.  H  F  ~~>v  x )
 
Theoremhlimeui 21766* The limit of a Hilbert space sequence is unique. (Contributed by NM, 19-Aug-1999.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ( E. x  F  ~~>v  x  <->  E! x  F  ~~>v  x )
 
Theoremisch3 21767* A Hilbert subspace is closed iff it is complete. A complete subspace is one in which every Cauchy sequence of vectors in the subspace converges to a member of the subspace (Definition of complete subspace in [Beran] p. 96). Remark 3.12 of [Beran] p. 107. (Contributed by NM, 24-Dec-2001.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ( H  e.  CH  <->  ( H  e.  SH  /\  A. f  e.  Cauchy  ( f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) ) )
 
Theoremchcompl 21768* Completeness of a closed subspace of Hilbert space. (Contributed by NM, 4-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  F  e.  Cauchy  /\  F : NN --> H )  ->  E. x  e.  H  F  ~~>v  x )
 
Theoremhelch 21769 The unit Hilbert lattice element (which is all of Hilbert space) belongs to the Hilbert lattice. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 6-Sep-1999.) (New usage is discouraged.)
 |-  ~H  e.  CH
 
Theoremhelsh 21770 Hilbert space is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  ~H  e.  SH
 
Theoremshsspwh 21771 Subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
 |-  SH  C_ 
 ~P ~H
 
Theoremchsspwh 21772 Closed subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
 |-  CH  C_  ~P ~H
 
Theoremhsn0elch 21773 The zero subspace belongs to the set of closed subspaces of Hilbert space. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
 |-  { 0h }  e.  CH
 
Theoremnorm1 21774 From any nonzero Hilbert space vector, construct a vector whose norm is 1. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  A  =/=  0h )  ->  ( normh `  ( (
 1  /  ( normh `  A ) )  .h  A ) )  =  1 )
 
Theoremnorm1exi 21775* A normalized vector exists in a subspace iff the subspace has a nonzero vector. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)
 |-  H  e.  SH   =>    |-  ( E. x  e.  H  x  =/=  0h  <->  E. y  e.  H  ( normh `  y )  =  1 )
 
Theoremnorm1hex 21776 A normalized vector can exist only iff the Hilbert space has a nonzero vector. (Contributed by NM, 21-Jan-2006.) (New usage is discouraged.)
 |-  ( E. x  e.  ~H  x  =/=  0h  <->  E. y  e.  ~H  ( normh `  y )  =  1 )
 
15.9.17  Orthocomplements
 
Definitiondf-oc 21777* Define orthogonal complement of a subset (usually a subspace) of Hilbert space. The orthogonal complement is the set of all vectors orthogonal to all vectors in the subset. See ocval 21805 and chocvali 21824 for its value. Textbooks usually denote this unary operation with the symbol  _|_ as a small superscript, although Mittelstaedt uses the symbol as a prefix operation. Here we define a function (prefix operation)  _|_ rather than introducing a new syntactical form. This lets us take advantage of the theorems about functions that we already have proved under set theory. Definition of [Mittelstaedt] p. 9. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)
 |-  _|_  =  ( x  e.  ~P ~H  |->  { y  e.  ~H  |  A. z  e.  x  ( y  .ih  z )  =  0 } )
 
Definitiondf-ch0 21778 Define the zero for closed subspaces of Hilbert space. See h0elch 21780 for closure law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
 |-  0H  =  { 0h }
 
Theoremelch0 21779 Membership in zero for closed subspaces of Hilbert space. (Contributed by NM, 6-Apr-2001.) (New usage is discouraged.)
 |-  ( A  e.  0H  <->  A  =  0h )
 
Theoremh0elch 21780 The zero subspace is a closed subspace. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
 |-  0H  e.  CH
 
Theoremh0elsh 21781 The zero subspace is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  0H  e.  SH
 
Theoremhhssva 21782 The vector addition operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   =>    |-  (  +h  |`  ( H  X.  H ) )  =  ( +v `  W )
 
Theoremhhsssm 21783 The scalar multiplication operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   =>    |-  (  .h  |`  ( CC 
 X.  H ) )  =  ( .s OLD `  W )
 
Theoremhhssnm 21784 The norm operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   =>    |-  ( normh  |`  H )  =  ( normCV `  W )
 
Theoremhhssabloi 21785 Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  H  e.  SH   =>    |-  (  +h  |`  ( H  X.  H ) )  e.  AbelOp
 
Theoremhhssablo 21786 Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)
 |-  ( H  e.  SH  ->  (  +h  |`  ( H  X.  H ) )  e.  AbelOp )
 
Theoremhhssnv 21787 Normed complex vector space property of a subspace. (Contributed by NM, 26-Mar-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   &    |-  H  e.  SH   =>    |-  W  e.  NrmCVec
 
Theoremhhssnvt 21788 Normed complex vector space property of a subspace. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   =>    |-  ( H  e.  SH  ->  W  e.  NrmCVec )
 
Theoremhhsst 21789 A member of  SH is a subspace. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) ) >. ,  ( normh  |`  H )
 >.   =>    |-  ( H  e.  SH  ->  W  e.  ( SubSp `  U ) )
 
Theoremhhshsslem1 21790 Lemma for hhsssh 21792. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) ) >. ,  ( normh  |`  H )
 >.   &    |-  W  e.  ( SubSp `  U )   &    |-  H  C_  ~H   =>    |-  H  =  (
 BaseSet `  W )
 
Theoremhhshsslem2 21791 Lemma for hhsssh 21792. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) ) >. ,  ( normh  |`  H )
 >.   &    |-  W  e.  ( SubSp `  U )   &    |-  H  C_  ~H   =>    |-  H  e.  SH
 
Theoremhhsssh 21792 The predicate " H is a subspace of Hilbert space." (Contributed by NM, 25-Mar-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) ) >. ,  ( normh  |`  H )
 >.   =>    |-  ( H  e.  SH  <->  ( W  e.  ( SubSp `  U )  /\  H  C_ 
 ~H ) )
 
Theoremhhsssh2 21793 The predicate " H is a subspace of Hilbert space." (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   =>    |-  ( H  e.  SH  <->  ( W  e.  NrmCVec  /\  H  C_  ~H ) )
 
Theoremhhssba 21794 The base set of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   &    |-  H  e.  SH   =>    |-  H  =  (
 BaseSet `  W )
 
Theoremhhssvs 21795 The vector subtraction operation on a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   &    |-  H  e.  SH   =>    |-  (  -h  |`  ( H  X.  H ) )  =  ( -v `  W )
 
Theoremhhssvsf 21796 Mapping of the vector subtraction operation on a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   &    |-  H  e.  SH   =>    |-  (  -h  |`  ( H  X.  H ) ) : ( H  X.  H ) --> H
 
Theoremhhssph 21797 Inner product space property of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   &    |-  H  e.  SH   =>    |-  W  e.  CPreHil OLD
 
Theoremhhssims 21798 Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   &    |-  H  e.  SH   &    |-  D  =  ( ( normh  o.  -h  )  |`  ( H  X.  H ) )   =>    |-  D  =  (
 IndMet `  W )
 
Theoremhhssims2 21799 Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   &    |-  D  =  ( IndMet `  W )   &    |-  H  e.  SH   =>    |-  D  =  ( ( normh  o.  -h  )  |`  ( H  X.  H ) )
 
Theoremhhssmet 21800 Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   &    |-  D  =  ( IndMet `  W )   &    |-  H  e.  SH   =>    |-  D  e.  ( Met `  H )
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