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Theorem List for Metamath Proof Explorer - 26301-26400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhelsh 26301 Hilbert space is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  ~H  e.  SH
 
Theoremshsspwh 26302 Subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
 |-  SH  C_ 
 ~P ~H
 
Theoremchsspwh 26303 Closed subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
 |-  CH  C_  ~P ~H
 
Theoremhsn0elch 26304 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 26305 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 26306* 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 26307 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 )
 
20.4.3  Orthocomplements
 
Definitiondf-oc 26308* 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 26336 and chocvali 26355 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 26309 Define the zero for closed subspaces of Hilbert space. See h0elch 26311 for closure law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
 |-  0H  =  { 0h }
 
Theoremelch0 26310 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 26311 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 26312 The zero subspace is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  0H  e.  SH
 
Theoremhhssva 26313 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 26314 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 ) )  =  ( .sOLD `  W )
 
Theoremhhssnm 26315 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 26316 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 26317 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 26318 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 26319 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 26320 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 26321 Lemma for hhsssh 26323. (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 26322 Lemma for hhsssh 26323. (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 26323 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 26324 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 26325 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 26326 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 26327 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 26328 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 26329 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 26330 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 26331 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 26332 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 26333 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 26334 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 26335 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.  CHilOLD
 
Theoremocval 26336* 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 26337* 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 26338* 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 26339 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 26340 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 26341 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 26342 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 26343 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 26344 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 26345 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 26346 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 26347 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 26348 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 26349 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 26350 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 26351 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 26352 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 26353 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 26354 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 26355* 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 26356 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 26357 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 26358* 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 26359 Lemma for occl 26360. (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 26360 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 26361 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 )
 
Theoremchoccl 26362 Closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( _|_ `  A )  e. 
 CH )
 
Theoremchoccli 26363 Closure of  CH orthocomplement. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( _|_ `  A )  e.  CH
 
20.4.4  Subspace sum, span, lattice join, lattice supremum
 
Definitiondf-shs 26364* Define subspace sum in  SH. See shsval 26368, shsval2i 26443, and shsval3i 26444 for its value. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)
 |-  +H  =  ( x  e.  SH ,  y  e.  SH  |->  (  +h  " ( x  X.  y ) ) )
 
Definitiondf-span 26365* Define the linear span of a subset of Hilbert space. Definition of span in [Schechter] p. 276. See spanval 26389 for its value. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  span  =  ( x  e.  ~P ~H  |->  |^| { y  e. 
 SH  |  x  C_  y } )
 
Definitiondf-chj 26366* Define Hilbert lattice join. See chjval 26408 for its value and chjcl 26413 for its closure law. Note that we define it over all Hilbert space subsets to allow proving more general theorems. Even for general subsets the join belongs to  CH; see sshjcl 26411. (Contributed by NM, 1-Nov-2000.) (New usage is discouraged.)
 |-  vH  =  ( x  e.  ~P ~H ,  y  e.  ~P ~H  |->  ( _|_ `  ( _|_ `  ( x  u.  y ) ) ) )
 
Definitiondf-chsup 26367 Define the supremum of a set of Hilbert lattice elements. See chsupval2 26466 for its value. We actually define the supremum for an arbitrary collection of Hilbert space subsets, not just elements of the Hilbert lattice  CH, to allow more general theorems. Even for general subsets the supremum still a Hilbert lattice element; see hsupcl 26395. (Contributed by NM, 9-Dec-2003.) (New usage is discouraged.)
 |-  \/H  =  ( x  e.  ~P ~P ~H  |->  ( _|_ `  ( _|_ `  U. x ) ) )
 
Theoremshsval 26368 Value of subspace sum of two Hilbert space subspaces. Definition of subspace sum in [Kalmbach] p. 65. (Contributed by NM, 16-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B )  =  (  +h  " ( A  X.  B ) ) )
 
Theoremshsss 26369 The subspace sum is a subset of Hilbert space. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B )  C_  ~H )
 
Theoremshsel 26370* Membership in the subspace sum of two Hilbert subspaces. (Contributed by NM, 14-Dec-2004.) (Revised by Mario Carneiro, 29-Jan-2014.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( C  e.  ( A  +H  B )  <->  E. x  e.  A  E. y  e.  B  C  =  ( x  +h  y ) ) )
 
Theoremshsel3 26371* Membership in the subspace sum of two Hilbert subspaces, using vector subtraction. (Contributed by NM, 20-Jan-2007.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( C  e.  ( A  +H  B )  <->  E. x  e.  A  E. y  e.  B  C  =  ( x  -h  y ) ) )
 
Theoremshseli 26372* Membership in subspace sum. (Contributed by NM, 4-May-2000.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( C  e.  ( A  +H  B )  <->  E. x  e.  A  E. y  e.  B  C  =  ( x  +h  y ) )
 
Theoremshscli 26373 Closure of subspace sum. (Contributed by NM, 15-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( A  +H  B )  e. 
 SH
 
Theoremshscl 26374 Closure of subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B )  e.  SH )
 
Theoremshscom 26375 Commutative law for subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B )  =  ( B  +H  A ) )
 
Theoremshsva 26376 Vector sum belongs to subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( ( C  e.  A  /\  D  e.  B )  ->  ( C  +h  D )  e.  ( A  +H  B ) ) )
 
Theoremshsel1 26377 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( C  e.  A  ->  C  e.  ( A  +H  B ) ) )
 
Theoremshsel2 26378 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( C  e.  B  ->  C  e.  ( A  +H  B ) ) )
 
Theoremshsvs 26379 Vector subtraction belongs to subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( ( C  e.  A  /\  D  e.  B )  ->  ( C  -h  D )  e.  ( A  +H  B ) ) )
 
Theoremshsub1 26380 Subspace sum is an upper bound of its arguments. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  A  C_  ( A  +H  B ) )
 
Theoremshsub2 26381 Subspace sum is an upper bound of its arguments. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  A  C_  ( B  +H  A ) )
 
Theoremchoc0 26382 The orthocomplement of the zero subspace is the unit subspace. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
 |-  ( _|_ `  0H )  =  ~H
 
Theoremchoc1 26383 The orthocomplement of the unit subspace is the zero subspace. Does not require Axiom of Choice. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
 |-  ( _|_ `  ~H )  =  0H
 
Theoremchocnul 26384 Orthogonal complement of the empty set. (Contributed by NM, 31-Oct-2000.) (New usage is discouraged.)
 |-  ( _|_ `  (/) )  =  ~H
 
Theoremshintcli 26385 Closure of intersection of a nonempty subset of  SH. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
 |-  ( A  C_  SH  /\  A  =/= 
 (/) )   =>    |- 
 |^| A  e.  SH
 
Theoremshintcl 26386 The intersection of a nonempty set of subspaces is a subspace. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  C_  SH  /\  A  =/=  (/) )  ->  |^| A  e.  SH )
 
Theoremchintcli 26387 The intersection of a nonempty set of closed subspaces is a closed subspace. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
 |-  ( A  C_  CH  /\  A  =/=  (/) )   =>    |- 
 |^| A  e.  CH
 
Theoremchintcl 26388 The intersection (infimum) of a nonempty subset of  CH belongs to  CH. Part of Theorem 3.13 of [Beran] p. 108. Also part of Definition 3.4-1 in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( A  C_  CH  /\  A  =/=  (/) )  ->  |^| A  e.  CH )
 
Theoremspanval 26389* Value of the linear span of a subset of Hilbert space. The span is the intersection of all subspaces constraining the subset. Definition of span in [Schechter] p. 276. (Contributed by NM, 2-Jun-2004.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  ( A  C_  ~H  ->  ( span `  A )  = 
 |^| { x  e.  SH  |  A  C_  x }
 )
 
Theoremhsupval 26390 Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 26465. (Contributed by NM, 9-Dec-2003.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  ( A  C_  ~P ~H  ->  ( 
 \/H  `  A )  =  ( _|_ `  ( _|_ `  U. A ) ) )
 
Theoremchsupval 26391 The value of the supremum of a set of closed subspaces of Hilbert space. For an alternate version of the value, see chsupval2 26466. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
 |-  ( A  C_  CH  ->  (  \/H  `  A )  =  ( _|_ `  ( _|_ ` 
 U. A ) ) )
 
Theoremspancl 26392 The span of a subset of Hilbert space is a subspace. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  ( A  C_  ~H  ->  ( span `  A )  e. 
 SH )
 
Theoremelspancl 26393 A member of a span is a vector. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  C_  ~H  /\  B  e.  ( span `  A ) )  ->  B  e.  ~H )
 
Theoremshsupcl 26394 Closure of the subspace supremum of set of subsets of Hilbert space. (Contributed by NM, 26-Nov-2004.) (New usage is discouraged.)
 |-  ( A  C_  ~P ~H  ->  (
 span `  U. A )  e.  SH )
 
Theoremhsupcl 26395 Closure of supremum of set of subsets of Hilbert space. Note that the supremum belongs to  CH even if the subsets do not. (Contributed by NM, 10-Nov-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  ( A  C_  ~P ~H  ->  ( 
 \/H  `  A )  e.  CH )
 
Theoremchsupcl 26396 Closure of supremum of subset of 
CH. Definition of supremum in Proposition 1 of [Kalmbach] p. 65. Shows that  CH is a complete lattice. Also part of Definition 3.4-1 in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 10-Nov-1999.) (New usage is discouraged.)
 |-  ( A  C_  CH  ->  (  \/H  `  A )  e.  CH )
 
Theoremhsupss 26397 Subset relation for supremum of Hilbert space subsets. (Contributed by NM, 24-Nov-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  (
 ( A  C_  ~P ~H  /\  B  C_  ~P ~H )  ->  ( A  C_  B  ->  (  \/H  `  A ) 
 C_  (  \/H  `  B ) ) )
 
Theoremchsupss 26398 Subset relation for supremum of subset of  CH. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
 |-  (
 ( A  C_  CH  /\  B  C_  CH )  ->  ( A  C_  B  ->  ( 
 \/H  `  A )  C_  (  \/H  `  B ) ) )
 
Theoremhsupunss 26399 The union of a set of Hilbert space subsets is smaller than its supremum. (Contributed by NM, 24-Nov-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  ( A  C_  ~P ~H  ->  U. A  C_  (  \/H  `  A ) )
 
Theoremchsupunss 26400 The union of a set of closed subspaces is smaller than its supremum. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
 |-  ( A  C_  CH  ->  U. A  C_  (  \/H  `  A ) )
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