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Theorem List for Metamath Proof Explorer - 21501-21600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhlmulass 21501 Hilbert space scalar multiplication associative law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X ) )  ->  ( ( A  x.  B ) S C )  =  ( A S ( B S C ) ) )
 
Theoremhldi 21502 Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  ( A S ( B G C ) )  =  ( ( A S B ) G ( A S C ) ) )
 
Theoremhldir 21503 Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  X ) )  ->  ( ( A  +  B ) S C )  =  ( ( A S C ) G ( B S C ) ) )
 
Theoremhlmul0 21504 Hilbert space scalar multiplication by zero. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  Z  =  ( 0vec `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  A  e.  X )  ->  ( 0 S A )  =  Z )
 
Theoremhlipf 21505 Mapping for Hilbert space inner product. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( U  e.  CHil OLD 
 ->  P : ( X  X.  X ) --> CC )
 
Theoremhlipcj 21506 Conjugate law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( A P B )  =  ( * `  ( B P A ) ) )
 
Theoremhlipdir 21507 Distributive law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( ( A G B ) P C )  =  ( ( A P C )  +  ( B P C ) ) )
 
Theoremhlipass 21508 Associative law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i
 OLD `  U )   =>    |-  (
 ( U  e.  CHil OLD  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A S B ) P C )  =  ( A  x.  ( B P C ) ) )
 
Theoremhlipgt0 21509 The inner product of a Hilbert space vector by itself is positive. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  CHil OLD  /\  A  e.  X  /\  A  =/=  Z ) 
 ->  0  <  ( A P A ) )
 
Theoremhlcompl 21510 Completeness of a Hilbert space. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
 |-  D  =  ( IndMet `  U )   &    |-  J  =  (
 MetOpen `  D )   =>    |-  ( ( U  e.  CHil OLD  /\  F  e.  ( Cau `  D )
 )  ->  F  e.  dom  ( ~~> t `  J ) )
 
16.6.3  Examples of complex Hilbert spaces
 
Theoremcnchl 21511 The set of complex numbers is a complex Hilbert space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +  ,  x.  >. ,  abs >.   =>    |-  U  e.  CHil OLD
 
16.6.4  Subspaces
 
Theoremssphl 21512 A Banach subspace of an inner product space is a Hilbert space. (Contributed by NM, 11-Apr-2008.) (New usage is discouraged.)
 |-  H  =  ( SubSp `  U )   =>    |-  ( ( U  e.  CPreHil OLD  /\  W  e.  H  /\  W  e.  CBan )  ->  W  e.  CHil OLD )
 
16.6.5  Hellinger-Toeplitz Theorem
 
Theoremhtthlem 21513* Lemma for htth 21514. The collection  K, which consists of functions  F ( z ) ( w )  =  <. w  |  T
( z ) >.  =  <. T ( w )  |  z >. for each  z in the unit ball, is a collection of bounded linear functions by ipblnfi 21450, so by the Uniform Boundedness theorem ubth 21468, there is a uniform bound  y on  ||  F ( x )  || for all  x in the unit ball. Then  |  T (
x )  |  ^
2  =  <. T ( x )  |  T
( x ) >.  =  F ( x ) (  T ( x ) )  <_  y  |  T ( x )  |, so  |  T ( x )  |  <_  y and 
T is bounded. (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  L  =  ( U 
 LnOp  U )   &    |-  B  =  ( U  BLnOp  U )   &    |-  N  =  ( normCV `  U )   &    |-  U  e.  CHil OLD   &    |-  W  =  <. <.  +  ,  x.  >. ,  abs >.   &    |-  ( ph  ->  T  e.  L )   &    |-  ( ph  ->  A. x  e.  X  A. y  e.  X  ( x P ( T `  y
 ) )  =  ( ( T `  x ) P y ) )   &    |-  F  =  ( z  e.  X  |->  ( w  e.  X  |->  ( w P ( T `  z
 ) ) ) )   &    |-  K  =  ( F " { z  e.  X  |  ( N `  z
 )  <_  1 }
 )   =>    |-  ( ph  ->  T  e.  B )
 
Theoremhtth 21514* Hellinger-Toeplitz Theorem: any self-adjoint linear operator defined on all of Hilbert space is bounded. Theorem 10.1-1 of [Kreyszig] p. 525. Discovered by E. Hellinger and O. Toeplitz in 1910, "it aroused both admiration and puzzlement since the theorem establishes a relation between properties of two different kinds, namely, the properties of being defined everywhere and being bounded." (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  L  =  ( U 
 LnOp  U )   &    |-  B  =  ( U  BLnOp  U )   =>    |-  ( ( U  e.  CHil OLD  /\  T  e.  L  /\  A. x  e.  X  A. y  e.  X  ( x P ( T `  y
 ) )  =  ( ( T `  x ) P y ) ) 
 ->  T  e.  B )
 
PART 17  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)

This part contains the definitions and theorems used by the Hilbert Space Explorer http://us.metamath.org/mpeuni/mmhil.html. Because it axiomatizes a single complex Hilbert space whose existence is assumed, its usefulness is limited. For example, it cannot work with real or quaternion Hilbert spaces and it cannot study relationships between two Hilbert spaces. More information can be found on the Hilbert Space Explorer page. Future development should work with general Hilbert spaces as defined by df-hil 16620.

 
17.1  Axiomatization of complex pre-Hilbert spaces
 
17.1.1  Basic Hilbert space definitions
 
Syntaxchil 21515 Extend class notation with Hilbert vector space.
 class  ~H
 
Syntaxcva 21516 Extend class notation with vector addition in Hilbert space. In the literature, the subscript "v" is omitted, but we need it to avoid ambiguity with complex number addition  + caddc 8756.
 class  +h
 
Syntaxcsm 21517 Extend class notation with scalar multiplication in Hilbert space. In the literature scalar multiplication is usually indicated by juxtaposition, but we need an explicit symbol to prevent ambiguity.
 class  .h
 
Syntaxcsp 21518 Extend class notation with inner (scalar) product in Hilbert space. In the literature, the inner product of  A and  B is usually written  <. A ,  B >. but our operation notation allows us to use existing theorems about operations and also eliminates ambiguity with the definition of an ordered pair df-op 3662.
 class  .ih
 
Syntaxcno 21519 Extend class notation with the norm function in Hilbert space. In the literature, the norm of  A is usually written "||  A ||", but we use function notation to take advantage of our existing theorems about functions.
 class  normh
 
Syntaxc0v 21520 Extend class notation with zero vector in Hilbert space.
 class  0h
 
Syntaxcmv 21521 Extend class notation with vector subtraction in Hilbert space.
 class  -h
 
Syntaxccau 21522 Extend class notation with set of Cauchy sequences in Hilbert space.
 class  Cauchy
 
Syntaxchli 21523 Extend class notation with convergence relation in Hilbert space.
 class  ~~>v
 
Syntaxcsh 21524 Extend class notation with set of subspaces of a Hilbert space.
 class  SH
 
Syntaxcch 21525 Extend class notation with set of closed subspaces of a Hilbert space.
 class  CH
 
Syntaxcort 21526 Extend class notation with orthogonal complement in  CH.
 class  _|_
 
Syntaxcph 21527 Extend class notation with subspace sum in  CH.
 class  +H
 
Syntaxcspn 21528 Extend class notation with subspace span in  CH.
 class  span
 
Syntaxchj 21529 Extend class notation with join in  CH.
 class  vH
 
Syntaxchsup 21530 Extend class notation with supremum of a collection in  CH.
 class  \/H
 
Syntaxc0h 21531 Extend class notation with zero of  CH.
 class  0H
 
Syntaxccm 21532 Extend class notation with the commutes relation on a Hilbert lattice.
 class  C_H
 
Syntaxcpjh 21533 Extend class notation with set of projections on a Hilbert space.
 class  proj  h
 
Syntaxchos 21534 Extend class notation with sum of Hilbert space operators.
 class  +op
 
Syntaxchot 21535 Extend class notation with scalar product of a Hilbert space operator.
 class  .op
 
Syntaxchod 21536 Extend class notation with difference of Hilbert space operators.
 class  -op
 
Syntaxchfs 21537 Extend class notation with sum of Hilbert space functionals.
 class  +fn
 
Syntaxchft 21538 Extend class notation with scalar product of Hilbert space functional.
 class  .fn
 
Syntaxch0o 21539 Extend class notation with the Hilbert space zero operator.
 class  0hop
 
Syntaxchio 21540 Extend class notation with Hilbert space identity operator.
 class  Iop
 
Syntaxcnop 21541 Extend class notation with the operator norm function.
 class  normop
 
Syntaxccop 21542 Extend class notation with set of continuous Hilbert space operators.
 class  ConOp
 
Syntaxclo 21543 Extend class notation with set of linear Hilbert space operators.
 class  LinOp
 
Syntaxcbo 21544 Extend class notation with set of bounded linear operators.
 class  BndLinOp
 
Syntaxcuo 21545 Extend class notation with set of unitary Hilbert space operators.
 class  UniOp
 
Syntaxcho 21546 Extend class notation with set of Hermitian Hilbert space operators.
 class  HrmOp
 
Syntaxcnmf 21547 Extend class notation with the functional norm function.
 class  normfn
 
Syntaxcnl 21548 Extend class notation with the functional nullspace function.
 class  null
 
Syntaxccnfn 21549 Extend class notation with set of continuous Hilbert space functionals.
 class  ConFn
 
Syntaxclf 21550 Extend class notation with set of linear Hilbert space functionals.
 class  LinFn
 
Syntaxcado 21551 Extend class notation with Hilbert space adjoint function.
 class  adjh
 
Syntaxcbr 21552 Extend class notation with the bra of a vector in Dirac bra-ket notation.
 class  bra
 
Syntaxck 21553 Extend class notation with the outer product of two vectors in Dirac bra-ket notation.
 class  ketbra
 
Syntaxcleo 21554 Extend class notation with positive operator ordering.
 class  <_op
 
Syntaxcei 21555 Extend class notation with Hilbert space eigenvector function.
 class  eigvec
 
Syntaxcel 21556 Extend class notation with Hilbert space eigenvalue function.
 class  eigval
 
Syntaxcspc 21557 Extend class notation with the spectrum of an operator.
 class  Lambda
 
Syntaxcst 21558 Extend class notation with set of states on a Hilbert lattice.
 class  States
 
Syntaxchst 21559 Extend class notation with set of Hilbert-space-valued states on a Hilbert lattice.
 class  CHStates
 
Syntaxccv 21560 Extend class notation with the covers relation on a Hilbert lattice.
 class  <oH
 
Syntaxcat 21561 Extend class notation with set of atoms on a Hilbert lattice.
 class HAtoms
 
Syntaxcmd 21562 Extend class notation with the modular pair relation on a Hilbert lattice.
 class  MH
 
Syntaxcdmd 21563 Extend class notation with the dual modular pair relation on a Hilbert lattice.
 class  MH*
 
17.1.2  Preliminary ZFC lemmas
 
Definitiondf-hnorm 21564 Define the function for the norm of a vector of Hilbert space. See normval 21719 for its value and normcl 21720 for its closure. Theorems norm-i-i 21728, norm-ii-i 21732, and norm-iii-i 21734 show it has the expected properties of a norm. In the literature, the norm of  A is usually written "||  A ||", but we use function notation to take advantage of our existing theorems about functions. Definition of norm in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
 |-  normh  =  ( x  e.  dom  dom  .ih  |->  ( sqr `  ( x  .ih  x ) ) )
 
Definitiondf-hba 21565 Define base set of Hilbert space, for use if we want to develop Hilbert space independently from the axioms (see comments in ax-hilex 21595). Note that  ~H is considered a primitive in the Hilbert space axioms below, and we don't use this definition outside of this section. This definition can be proved independently from those axioms as theorem hhba 21762. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
 
Definitiondf-h0v 21566 Define the zero vector of Hilbert space. Note that  0vec is considered a primitive in the Hilbert space axioms below, and we don't use this definition outside of this section. It is proved from the axioms as theorem hh0v 21763. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  0h  =  ( 0vec `  <. <.  +h  ,  .h  >. ,  normh >. )
 
Definitiondf-hvsub 21567* Define vector subtraction. See hvsubvali 21616 for its value and hvsubcli 21617 for its closure. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
 |-  -h  =  ( x  e.  ~H ,  y  e.  ~H  |->  ( x  +h  ( -u 1  .h  y ) ) )
 
Definitiondf-hlim 21568* Define the limit relation for Hilbert space. See hlimi 21783 for its relational expression. Note that  f : NN --> ~H is an infinite sequence of vectors, i.e. a mapping from integers to vectors. Definition of converge in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
 |-  ~~>v  =  { <. f ,  w >.  |  ( ( f : NN --> ~H  /\  w  e. 
 ~H )  /\  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
 ( normh `  ( (
 f `  z )  -h  w ) )  < 
 x ) }
 
Definitiondf-hcau 21569* Define the set of Cauchy sequences on a Hilbert space. See hcau 21779 for its membership relation. Note that  f : NN --> ~H is an infinite sequence of vectors, i.e. a mapping from integers to vectors. Definition of Cauchy sequence in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
 |-  Cauchy  =  {
 f  e.  ( ~H 
 ^m  NN )  |  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
 ( normh `  ( (
 f `  y )  -h  ( f `  z
 ) ) )  < 
 x }
 
Theoremh2hva 21570 The group (addition) operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  NrmCVec   =>    |- 
 +h  =  ( +v
 `  U )
 
Theoremh2hsm 21571 The scalar product operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  NrmCVec   =>    |- 
 .h  =  ( .s
 OLD `  U )
 
Theoremh2hnm 21572 The norm function of Hilbert space. (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  NrmCVec   =>    |- 
 normh  =  ( normCV `  U )
 
Theoremh2hvs 21573 The vector subtraction operation of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  NrmCVec   &    |- 
 ~H  =  ( BaseSet `  U )   =>    |- 
 -h  =  ( -v
 `  U )
 
Theoremh2hmetdval 21574 Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  NrmCVec   &    |- 
 ~H  =  ( BaseSet `  U )   &    |-  D  =  (
 IndMet `  U )   =>    |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A D B )  =  ( normh `  ( A  -h  B ) ) )
 
Theoremh2hcau 21575 The Cauchy sequences of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  NrmCVec   &    |- 
 ~H  =  ( BaseSet `  U )   &    |-  D  =  (
 IndMet `  U )   =>    |-  Cauchy  =  ( ( Cau `  D )  i^i  ( ~H  ^m  NN ) )
 
Theoremh2hlm 21576 The limit sequences of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  NrmCVec   &    |- 
 ~H  =  ( BaseSet `  U )   &    |-  D  =  (
 IndMet `  U )   &    |-  J  =  ( MetOpen `  D )   =>    |-  ~~>v  =  ( ( ~~> t `  J )  |`  ( ~H  ^m  NN ) )
 
17.1.3  Derive the Hilbert space axioms from ZFC set theory

Before introducing the 18 axioms for Hilbert space, we first prove them as the conclusions of theorems axhilex-zf 21577 through axhcompl-zf 21594, using ZFC set theory only. These show that if we are given a known, fixed Hilbert space  U  =  <. <.  +h  ,  .h  >. ,  normh >. that satisfies their hypotheses, then we can derive the Hilbert space axioms as theorems of ZFC set theory. In practice, in order to use these theorems to convert the Hilbert Space explorer to a ZFC-only subtheory, we would also have to provide definitions for the 3 (otherwise primitive) class constants  +h,  .h, and  .ih before df-hnorm 21564 above. See also the comment in ax-hilex 21595.

 
Theoremaxhilex-zf 21577 Derive axiom ax-hilex 21595 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |- 
 ~H  e.  _V
 
Theoremaxhfvadd-zf 21578 Derive axiom ax-hfvadd 21596 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |- 
 +h  : ( ~H 
 X.  ~H ) --> ~H
 
Theoremaxhvcom-zf 21579 Derive axiom ax-hvcom 21597 from Hilbert space under ZF set theory. (Contributed by NM, 27-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |-  ( ( A  e.  ~H 
 /\  B  e.  ~H )  ->  ( A  +h  B )  =  ( B  +h  A ) )
 
Theoremaxhvass-zf 21580 Derive axiom ax-hvass 21598 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |-  ( ( A  e.  ~H 
 /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  +h  B )  +h  C )  =  ( A  +h  ( B  +h  C ) ) )
 
Theoremaxhv0cl-zf 21581 Derive axiom ax-hv0cl 21599 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |- 
 0h  e.  ~H
 
Theoremaxhvaddid-zf 21582 Derive axiom ax-hvaddid 21600 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |-  ( A  e.  ~H  ->  ( A  +h  0h )  =  A )
 
Theoremaxhfvmul-zf 21583 Derive axiom ax-hfvmul 21601 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |- 
 .h  : ( CC 
 X.  ~H ) --> ~H
 
Theoremaxhvmulid-zf 21584 Derive axiom ax-hvmulid 21602 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |-  ( A  e.  ~H  ->  ( 1  .h  A )  =  A )
 
Theoremaxhvmulass-zf 21585 Derive axiom ax-hvmulass 21603 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( ( A  x.  B )  .h  C )  =  ( A  .h  ( B  .h  C ) ) )
 
Theoremaxhvdistr1-zf 21586 Derive axiom ax-hvdistr1 21604 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .h  ( B  +h  C ) )  =  ( ( A  .h  B )  +h  ( A  .h  C ) ) )
 
Theoremaxhvdistr2-zf 21587 Derive axiom ax-hvdistr2 21605 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( ( A  +  B )  .h  C )  =  ( ( A  .h  C )  +h  ( B  .h  C ) ) )
 
Theoremaxhvmul0-zf 21588 Derive axiom ax-hvmul0 21606 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |-  ( A  e.  ~H  ->  ( 0  .h  A )  =  0h )
 
Theoremaxhfi-zf 21589 Derive axiom ax-hfi 21674 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   &    |- 
 .ih  =  ( .i OLD `  U )   =>    |-  .ih  : ( ~H  X.  ~H ) --> CC
 
Theoremaxhis1-zf 21590 Derive axiom ax-his1 21677 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   &    |- 
 .ih  =  ( .i OLD `  U )   =>    |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  B )  =  ( * `  ( B  .ih  A ) ) )
 
Theoremaxhis2-zf 21591 Derive axiom ax-his2 21678 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   &    |- 
 .ih  =  ( .i OLD `  U )   =>    |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  +h  B )  .ih  C )  =  ( ( A  .ih  C )  +  ( B  .ih  C ) ) )
 
Theoremaxhis3-zf 21592 Derive axiom ax-his3 21679 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   &    |- 
 .ih  =  ( .i OLD `  U )   =>    |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  .h  B )  .ih  C )  =  ( A  x.  ( B  .ih  C ) ) )
 
Theoremaxhis4-zf 21593 Derive axiom ax-his4 21680 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   &    |- 
 .ih  =  ( .i OLD `  U )   =>    |-  ( ( A  e.  ~H  /\  A  =/=  0h )  ->  0  <  ( A  .ih  A ) )
 
Theoremaxhcompl-zf 21594* Derive axiom ax-hcompl 21797 from Hilbert space under ZF set theory. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  U  e.  CHil OLD   =>    |-  ( F  e.  Cauchy  ->  E. x  e.  ~H  F  ~~>v  x )
 
17.1.4  Introduce the vector space axioms for a Hilbert space

Here we introduce the axioms a complex Hilbert space, which is the foundation for quantum mechanics and quantum field theory. The 18 axioms for a complex Hilbert space consist of ax-hilex 21595, ax-hfvadd 21596, ax-hvcom 21597, ax-hvass 21598, ax-hv0cl 21599, ax-hvaddid 21600, ax-hfvmul 21601, ax-hvmulid 21602, ax-hvmulass 21603, ax-hvdistr1 21604, ax-hvdistr2 21605, ax-hvmul0 21606, ax-hfi 21674, ax-his1 21677, ax-his2 21678, ax-his3 21679, ax-his4 21680, and ax-hcompl 21797.

The axioms specify the properties of 5 primitive symbols,  ~H,  +h,  .h,  0h, and  .ih.

If we can prove in ZFC set theory that a class  U  =  <. <.  +h  ,  .h  >. ,  normh >. is a complex Hilbert space, i.e. that  U  e.  CHil
OLD, then these axioms can be proved as theorems axhilex-zf 21577, axhfvadd-zf 21578, axhvcom-zf 21579, axhvass-zf 21580, axhv0cl-zf 21581, axhvaddid-zf 21582, axhfvmul-zf 21583, axhvmulid-zf 21584, axhvmulass-zf 21585, axhvdistr1-zf 21586, axhvdistr2-zf 21587, axhvmul0-zf 21588, axhfi-zf 21589, axhis1-zf 21590, axhis2-zf 21591, axhis3-zf 21592, axhis4-zf 21593, and axhcompl-zf 21594 respectively. In that case, the theorems of the Hilbert Space Explorer will become theorems of ZFC set theory. See also the comments in axhilex-zf 21577.

 
Axiomax-hilex 21595 This is our first axiom for a complex Hilbert space, which is the foundation for quantum mechanics and quantum field theory. We assume that there exists a primitive class,  ~H, which contains objects called vectors. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
 |-  ~H  e.  _V
 
Axiomax-hfvadd 21596 Vector addition is an operation on 
~H. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
 |-  +h  : ( ~H  X.  ~H )
 --> ~H
 
Axiomax-hvcom 21597 Vector addition is commutative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  B )  =  ( B  +h  A ) )
 
Axiomax-hvass 21598 Vector addition is associative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  +h  B )  +h  C )  =  ( A  +h  ( B  +h  C ) ) )
 
Axiomax-hv0cl 21599 The zero vector is in the vector space. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
 |-  0h  e.  ~H
 
Axiomax-hvaddid 21600 Addition with the zero vector. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( A  +h  0h )  =  A )
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