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Theorem List for Metamath Proof Explorer - 21801-21900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhilhl 21801 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 21802 Define the set of subspaces of a Hilbert space. See issh 21803 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 21803 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 21804* 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 21805 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 21806 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 21807 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 21808 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 21809 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 21810 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 21811 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 21812 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 21813 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 21814 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 21815* 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 21816 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 21817 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 21818. From Definition of [Beran] p. 107. Alternate definitions are given by isch2 21819 and isch3 21837. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)
 |-  CH  =  { h  e.  SH  |  (  ~~>v  " ( h  ^m  NN ) ) 
 C_  h }
 
Theoremisch 21818 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 21819* 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 21820 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 21821 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 21822 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 21823 A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  H  e.  SH
 
Theoremch0 21824 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 21825 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 21826 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 21827 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 21828 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 21829 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 21830 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 21831 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 21832 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 21833 Function-like behavior of the convergence relation. (Contributed by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ~~>v  : dom  ~~>v  --> ~H
 
Theoremhlimuni 21834 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 21835* 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 21836* 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 21837* 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 21838* 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 21839 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 21840 Hilbert space is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  ~H  e.  SH
 
Theoremshsspwh 21841 Subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
 |-  SH  C_ 
 ~P ~H
 
Theoremchsspwh 21842 Closed subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
 |-  CH  C_  ~P ~H
 
Theoremhsn0elch 21843 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 21844 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 21845* 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 21846 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 )
 
17.4.3  Orthocomplements
 
Definitiondf-oc 21847* 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 21875 and chocvali 21894 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 21848 Define the zero for closed subspaces of Hilbert space. See h0elch 21850 for closure law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
 |-  0H  =  { 0h }
 
Theoremelch0 21849 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 21850 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 21851 The zero subspace is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  0H  e.  SH
 
Theoremhhssva 21852 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 21853 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 21854 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 21855 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 21856 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 21857 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 21858 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 21859 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 21860 Lemma for hhsssh 21862. (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 21861 Lemma for hhsssh 21862. (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 21862 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 21863 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 21864 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 21865 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 21866 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 21867 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 21868 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 21869 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 21870 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 )
 
Theoremhhssmetdval 21871 Value of the distance function of the metric space 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   =>    |-  (
 ( A  e.  H  /\  B  e.  H ) 
 ->  ( A D B )  =  ( normh `  ( A  -h  B ) ) )
 
Theoremhhsscms 21872 The induced metric of a closed subspace is complete. (Contributed by NM, 10-Apr-2008.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   &    |-  D  =  ( IndMet `  W )   &    |-  H  e.  CH   =>    |-  D  e.  ( CMet `  H )
 
Theoremhhssbn 21873 Banach space property of a closed subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   &    |-  H  e.  CH   =>    |-  W  e.  CBan
 
Theoremhhsshl 21874 Hilbert space property of a closed subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  W  =  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC 
 X.  H ) )
 >. ,  ( normh  |`  H )
 >.   &    |-  H  e.  CH   =>    |-  W  e.  CHil OLD
 
Theoremocval 21875* Value of orthogonal complement of a subset of Hilbert space. (Contributed by NM, 7-Aug-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  ( H  C_  ~H  ->  ( _|_ `  H )  =  { x  e.  ~H  |  A. y  e.  H  ( x  .ih  y )  =  0 } )
 
Theoremocel 21876* Membership in orthogonal complement of H subset. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)
 |-  ( H  C_  ~H  ->  ( A  e.  ( _|_ `  H )  <->  ( A  e.  ~H 
 /\  A. x  e.  H  ( A  .ih  x )  =  0 ) ) )
 
Theoremshocel 21877* Membership in orthogonal complement of H subspace. (Contributed by NM, 9-Oct-1999.) (New usage is discouraged.)
 |-  ( H  e.  SH  ->  ( A  e.  ( _|_ `  H )  <->  ( A  e.  ~H 
 /\  A. x  e.  H  ( A  .ih  x )  =  0 ) ) )
 
Theoremocsh 21878 The orthogonal complement of a subspace is a subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)
 |-  ( A  C_  ~H  ->  ( _|_ `  A )  e. 
 SH )
 
Theoremshocsh 21879 The orthogonal complement of a subspace is a subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)
 |-  ( A  e.  SH  ->  ( _|_ `  A )  e.  SH )
 
Theoremocss 21880 An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
 |-  ( A  C_  ~H  ->  ( _|_ `  A )  C_  ~H )
 
Theoremshocss 21881 An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
 |-  ( A  e.  SH  ->  ( _|_ `  A )  C_ 
 ~H )
 
Theoremoccon 21882 Contraposition law for orthogonal complement. (Contributed by NM, 8-Aug-2000.) (New usage is discouraged.)
 |-  (
 ( A  C_  ~H  /\  B  C_  ~H )  ->  ( A  C_  B  ->  ( _|_ `  B )  C_  ( _|_ `  A ) ) )
 
Theoremoccon2 21883 Double contraposition for orthogonal complement. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
 |-  (
 ( A  C_  ~H  /\  B  C_  ~H )  ->  ( A  C_  B  ->  ( _|_ `  ( _|_ `  A ) )  C_  ( _|_ `  ( _|_ `  B ) ) ) )
 
Theoremoccon2i 21884 Double contraposition for orthogonal complement. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
 |-  A  C_ 
 ~H   &    |-  B  C_  ~H   =>    |-  ( A  C_  B  ->  ( _|_ `  ( _|_ `  A ) ) 
 C_  ( _|_ `  ( _|_ `  B ) ) )
 
Theoremoc0 21885 The zero vector belongs to an orthogonal complement of a Hilbert subspace. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
 |-  ( H  e.  SH  ->  0h  e.  ( _|_ `  H ) )
 
Theoremocorth 21886 Members of a subset and its complement are orthogonal. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
 |-  ( H  C_  ~H  ->  (
 ( A  e.  H  /\  B  e.  ( _|_ `  H ) )  ->  ( A  .ih  B )  =  0 ) )
 
Theoremshocorth 21887 Members of a subspace and its complement are orthogonal. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)
 |-  ( H  e.  SH  ->  ( ( A  e.  H  /\  B  e.  ( _|_ `  H ) )  ->  ( A  .ih  B )  =  0 ) )
 
Theoremococss 21888 Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
 |-  ( A  C_  ~H  ->  A  C_  ( _|_ `  ( _|_ `  A ) ) )
 
Theoremshococss 21889 Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)
 |-  ( A  e.  SH  ->  A 
 C_  ( _|_ `  ( _|_ `  A ) ) )
 
Theoremshorth 21890 Members of orthogonal subspaces are orthogonal. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
 |-  ( H  e.  SH  ->  ( G  C_  ( _|_ `  H )  ->  (
 ( A  e.  G  /\  B  e.  H ) 
 ->  ( A  .ih  B )  =  0 )
 ) )
 
Theoremocin 21891 Intersection of a Hilbert subspace and its complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
 |-  ( A  e.  SH  ->  ( A  i^i  ( _|_ `  A ) )  =  0H )
 
Theoremoccon3 21892 Hilbert lattice contraposition law. (Contributed by Mario Carneiro, 18-May-2014.) (New usage is discouraged.)
 |-  (
 ( A  C_  ~H  /\  B  C_  ~H )  ->  ( A  C_  ( _|_ `  B )  <->  B  C_  ( _|_ `  A ) ) )
 
Theoremocnel 21893 A nonzero vector in the complement of a subspace does not belong to the subspace. (Contributed by NM, 10-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( H  e.  SH  /\  A  e.  ( _|_ `  H )  /\  A  =/=  0h )  ->  -.  A  e.  H )
 
Theoremchocvali 21894* Value of the orthogonal complement of a Hilbert lattice element. The orthogonal complement of  A is the set of vectors that are orthogonal to all vectors in  A. (Contributed by NM, 8-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( _|_ `  A )  =  { x  e.  ~H  |  A. y  e.  A  ( x  .ih  y )  =  0 }
 
Theoremshuni 21895 Two subspaces with trivial intersection have a unique decomposition of the elements of the subspace sum. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  H  e.  SH )   &    |-  ( ph  ->  K  e.  SH )   &    |-  ( ph  ->  ( H  i^i  K )  =  0H )   &    |-  ( ph  ->  A  e.  H )   &    |-  ( ph  ->  B  e.  K )   &    |-  ( ph  ->  C  e.  H )   &    |-  ( ph  ->  D  e.  K )   &    |-  ( ph  ->  ( A  +h  B )  =  ( C  +h  D ) )   =>    |-  ( ph  ->  ( A  =  C  /\  B  =  D )
 )
 
Theoremchocunii 21896 Lemma for uniqueness part of Projection Theorem. Theorem 3.7(i) of [Beran] p. 102 (uniqueness part). (Contributed by NM, 23-Oct-1999.) (Proof shortened by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( ( ( A  e.  H  /\  B  e.  ( _|_ `  H ) )  /\  ( C  e.  H  /\  D  e.  ( _|_ `  H ) ) )  ->  ( ( R  =  ( A  +h  B ) 
 /\  R  =  ( C  +h  D ) )  ->  ( A  =  C  /\  B  =  D ) ) )
 
Theorempjhthmo 21897* Projection Theorem, uniqueness part. Any two disjoint subspaces yield a unique decomposition of vectors into each subspace. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH  /\  ( A  i^i  B )  =  0H )  ->  E* x ( x  e.  A  /\  E. y  e.  B  C  =  ( x  +h  y ) ) )
 
Theoremoccllem 21898 Lemma for occl 21899. (Contributed by NM, 7-Aug-2000.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  A  C_  ~H )   &    |-  ( ph  ->  F  e.  Cauchy )   &    |-  ( ph  ->  F : NN
 --> ( _|_ `  A ) )   &    |-  ( ph  ->  B  e.  A )   =>    |-  ( ph  ->  ( (  ~~>v  `  F )  .ih  B )  =  0 )
 
Theoremoccl 21899 Closure of complement of Hilbert subset. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 8-Aug-2000.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ( A  C_  ~H  ->  ( _|_ `  A )  e. 
 CH )
 
Theoremshoccl 21900 Closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 13-Oct-1999.) (New usage is discouraged.)
 |-  ( A  e.  SH  ->  ( _|_ `  A )  e.  CH )
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