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Theorem List for Metamath Proof Explorer - 27501-27600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
20.6.7  Dirac bra-ket notation
 
Definitiondf-bra 27501* Define the bra of a vector used by Dirac notation. Based on definition of bra in [Prugovecki] p. 186 (p. 180 in 1971 edition). In Dirac bra-ket notation,  <. A  |  B >. is a complex number equal to the inner product  ( B  .ih  A ). But physicists like to talk about the individual components 
<. A  | and  |  B >., called bra and ket respectively. In order for their properties to make sense formally, we define the ket  |  B >. as the vector  B itself, and the bra  <. A  | as a functional from  ~H to  CC. We represent the Dirac notation  <. A  |  B >. by  ( ( bra `  A
) `  B ); see braval 27595. The reversal of the inner product arguments not only makes the bra-ket behavior consistent with physics literature (see comments under ax-his3 26735) but is also required in order for the associative law kbass2 27768 to work.

Our definition of bra and the associated outer product df-kb 27502 differs from, but is equivalent to, a common approach in the literature that makes use of mappings to a dual space. Our approach eliminates the need to have a parallel development of this dual space and instead keeps everything in Hilbert space.

For an extensive discussion about how our notation maps to the bra-ket notation in physics textbooks, see http://us.metamath.org/mpeuni/mmnotes.txt, under the 17-May-2006 entry. (Contributed by NM, 15-May-2006.) (New usage is discouraged.)

 |-  bra  =  ( x  e.  ~H  |->  ( y  e.  ~H  |->  ( y  .ih  x ) ) )
 
Definitiondf-kb 27502* Define a commuted bra and ket juxtaposition used by Dirac notation. In Dirac notation,  |  A >.  <. B  | is an operator known as the outer product of  A and  B, which we represent by  ( A  ketbra  B ). Based on Equation 8.1 of [Prugovecki] p. 376. This definition, combined with definition df-bra 27501, allows any legal juxtaposition of bras and kets to make sense formally and also to obey the associative law when mapped back to Dirac notation. (Contributed by NM, 15-May-2006.) (New usage is discouraged.)
 |-  ketbra  =  ( x  e.  ~H ,  y  e.  ~H  |->  ( z  e.  ~H  |->  ( ( z  .ih  y )  .h  x ) ) )
 
20.6.8  Positive operators
 
Definitiondf-leop 27503* Define positive operator ordering. Definition VI.1 of [Retherford] p. 49. Note that  ( ~H  X.  0H )  <_op  T means that  T is a positive operator. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
 |-  <_op  =  { <. t ,  u >.  |  ( ( u 
 -op  t )  e. 
 HrmOp  /\  A. x  e. 
 ~H  0  <_  (
 ( ( u  -op  t ) `  x )  .ih  x ) ) }
 
20.6.9  Eigenvectors, eigenvalues, spectrum
 
Definitiondf-eigvec 27504* Define the eigenvector function. Theorem eleigveccl 27610 shows that  eigvec `  T, the set of eigenvectors of Hilbert space operator  T, are Hilbert space vectors. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
 |-  eigvec  =  ( t  e.  ( ~H 
 ^m  ~H )  |->  { x  e.  ( ~H  \  0H )  |  E. z  e.  CC  ( t `  x )  =  (
 z  .h  x ) } )
 
Definitiondf-eigval 27505* Define the eigenvalue function. The range of  eigval `  T is the set of eigenvalues of Hilbert space operator  T. Theorem eigvalcl 27612 shows that  ( eigval `  T
) `  A, the eigenvalue associated with eigenvector  A, is a complex number. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
 |-  eigval  =  ( t  e.  ( ~H 
 ^m  ~H )  |->  ( x  e.  ( eigvec `  t
 )  |->  ( ( ( t `  x ) 
 .ih  x )  /  ( ( normh `  x ) ^ 2 ) ) ) )
 
Definitiondf-spec 27506* Define the spectrum of an operator. Definition of spectrum in [Halmos] p. 50. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
 |-  Lambda  =  ( t  e.  ( ~H 
 ^m  ~H )  |->  { x  e.  CC  |  -.  (
 t  -op  ( x  .op  (  _I  |`  ~H )
 ) ) : ~H -1-1-> ~H
 } )
 
20.6.10  Theorems about operators and functionals
 
Theoremnmopval 27507* Value of the norm of a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  ( normop `
  T )  = 
 sup ( { x  |  E. y  e.  ~H  ( ( normh `  y
 )  <_  1  /\  x  =  ( normh `  ( T `  y
 ) ) ) } ,  RR* ,  <  )
 )
 
Theoremelcnop 27508* Property defining a continuous Hilbert space operator. (Contributed by NM, 28-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  ( T  e.  ConOp  <->  ( T : ~H
 --> ~H  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  (
 ( normh `  ( w  -h  x ) )  < 
 z  ->  ( normh `  ( ( T `  w )  -h  ( T `  x ) ) )  <  y ) ) )
 
Theoremellnop 27509* Property defining a linear Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  ( T  e.  LinOp  <->  ( T : ~H
 --> ~H  /\  A. x  e.  CC  A. y  e. 
 ~H  A. z  e.  ~H  ( T `  ( ( x  .h  y )  +h  z ) )  =  ( ( x  .h  ( T `  y ) )  +h  ( T `  z ) ) ) )
 
Theoremlnopf 27510 A linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
 |-  ( T  e.  LinOp  ->  T : ~H --> ~H )
 
Theoremelbdop 27511 Property defining a bounded linear Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  ( T  e.  BndLinOp  <->  ( T  e.  LinOp  /\  ( normop `  T )  < +oo ) )
 
Theorembdopln 27512 A bounded linear Hilbert space operator is a linear operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  BndLinOp  ->  T  e.  LinOp
 )
 
Theorembdopf 27513 A bounded linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  BndLinOp  ->  T : ~H
 --> ~H )
 
TheoremnmopsetretALT 27514* The set in the supremum of the operator norm definition df-nmop 27490 is a set of reals. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( T : ~H --> ~H  ->  { x  |  E. y  e.  ~H  ( ( normh `  y )  <_  1  /\  x  =  ( normh `  ( T `  y ) ) ) }  C_  RR )
 
TheoremnmopsetretHIL 27515* The set in the supremum of the operator norm definition df-nmop 27490 is a set of reals. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  { x  |  E. y  e.  ~H  ( ( normh `  y )  <_  1  /\  x  =  ( normh `  ( T `  y ) ) ) }  C_  RR )
 
Theoremnmopsetn0 27516* The set in the supremum of the operator norm definition df-nmop 27490 is nonempty. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
 |-  ( normh `  ( T `  0h ) )  e.  { x  |  E. y  e.  ~H  ( ( normh `  y )  <_  1  /\  x  =  ( normh `  ( T `  y ) ) ) }
 
Theoremnmopxr 27517 The norm of a Hilbert space operator is an extended real. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  ( normop `
  T )  e.  RR* )
 
Theoremnmoprepnf 27518 The norm of a Hilbert space operator is either real or plus infinity. (Contributed by NM, 5-Feb-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  ( ( normop `  T )  e.  RR  <->  ( normop `  T )  =/= +oo ) )
 
Theoremnmopgtmnf 27519 The norm of a Hilbert space operator is not minus infinity. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  -> -oo 
 <  ( normop `  T )
 )
 
Theoremnmopreltpnf 27520 The norm of a Hilbert space operator is real iff it is less than infinity. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  ( ( normop `  T )  e.  RR  <->  ( normop `  T )  < +oo ) )
 
Theoremnmopre 27521 The norm of a bounded operator is a real number. (Contributed by NM, 29-Jan-2006.) (New usage is discouraged.)
 |-  ( T  e.  BndLinOp  ->  ( normop `  T )  e.  RR )
 
Theoremelbdop2 27522 Property defining a bounded linear Hilbert space operator. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  BndLinOp  <->  ( T  e.  LinOp  /\  ( normop `  T )  e.  RR ) )
 
Theoremelunop 27523* Property defining a unitary Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
 |-  ( T  e.  UniOp  <->  ( T : ~H -onto-> ~H  /\  A. x  e.  ~H  A. y  e. 
 ~H  ( ( T `
  x )  .ih  ( T `  y ) )  =  ( x 
 .ih  y ) ) )
 
Theoremelhmop 27524* Property defining a Hermitian Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  ( T  e.  HrmOp  <->  ( T : ~H
 --> ~H  /\  A. x  e.  ~H  A. y  e. 
 ~H  ( x  .ih  ( T `  y ) )  =  ( ( T `  x ) 
 .ih  y ) ) )
 
Theoremhmopf 27525 A Hermitian operator is a Hilbert space operator (mapping). (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)
 |-  ( T  e.  HrmOp  ->  T : ~H --> ~H )
 
Theoremhmopex 27526 The class of Hermitian operators is a set. (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
 |-  HrmOp  e.  _V
 
Theoremnmfnval 27527* Value of the norm of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  ( T : ~H --> CC  ->  (
 normfn `  T )  = 
 sup ( { x  |  E. y  e.  ~H  ( ( normh `  y
 )  <_  1  /\  x  =  ( abs `  ( T `  y
 ) ) ) } ,  RR* ,  <  )
 )
 
Theoremnmfnsetre 27528* The set in the supremum of the functional norm definition df-nmfn 27496 is a set of reals. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> CC  ->  { x  |  E. y  e.  ~H  ( ( normh `  y )  <_  1  /\  x  =  ( abs `  ( T `  y ) ) ) }  C_  RR )
 
Theoremnmfnsetn0 27529* The set in the supremum of the functional norm definition df-nmfn 27496 is nonempty. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
 |-  ( abs `  ( T `  0h ) )  e.  { x  |  E. y  e.  ~H  ( ( normh `  y )  <_  1  /\  x  =  ( abs `  ( T `  y ) ) ) }
 
Theoremnmfnxr 27530 The norm of any Hilbert space functional is an extended real. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> CC  ->  (
 normfn `  T )  e.  RR* )
 
Theoremnmfnrepnf 27531 The norm of a Hilbert space functional is either real or plus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
 |-  ( T : ~H --> CC  ->  ( ( normfn `  T )  e.  RR  <->  ( normfn `  T )  =/= +oo ) )
 
Theoremnlfnval 27532 Value of the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> CC  ->  (
 null `  T )  =  ( `' T " { 0 } )
 )
 
Theoremelcnfn 27533* Property defining a continuous functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  ( T  e.  ConFn  <->  ( T : ~H
 --> CC  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  (
 ( normh `  ( w  -h  x ) )  < 
 z  ->  ( abs `  ( ( T `  w )  -  ( T `  x ) ) )  <  y ) ) )
 
Theoremellnfn 27534* Property defining a linear functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  ( T  e.  LinFn  <->  ( T : ~H
 --> CC  /\  A. x  e.  CC  A. y  e. 
 ~H  A. z  e.  ~H  ( T `  ( ( x  .h  y )  +h  z ) )  =  ( ( x  x.  ( T `  y ) )  +  ( T `  z ) ) ) )
 
Theoremlnfnf 27535 A linear Hilbert space functional is a functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
 |-  ( T  e.  LinFn  ->  T : ~H --> CC )
 
Theoremdfadj2 27536* Alternate definition of the adjoint of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
 |-  adjh  =  { <. t ,  u >.  |  ( t : ~H --> ~H  /\  u : ~H
 --> ~H  /\  A. x  e.  ~H  A. y  e. 
 ~H  ( x  .ih  ( t `  y
 ) )  =  ( ( u `  x )  .ih  y ) ) }
 
Theoremfunadj 27537 Functionality of the adjoint function. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
 |-  Fun  adjh
 
Theoremdmadjss 27538 The domain of the adjoint function is a subset of the maps from  ~H to  ~H. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
 |-  dom  adjh  C_  ( ~H  ^m  ~H )
 
Theoremdmadjop 27539 A member of the domain of the adjoint function is a Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  dom  adjh  ->  T : ~H --> ~H )
 
Theoremadjeu 27540* Elementhood in the domain of the adjoint function. (Contributed by Mario Carneiro, 11-Sep-2015.) (Revised by Mario Carneiro, 24-Dec-2016.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  ( T  e.  dom  adjh  <->  E! u  e.  ( ~H  ^m 
 ~H ) A. x  e.  ~H  A. y  e. 
 ~H  ( x  .ih  ( T `  y ) )  =  ( ( u `  x ) 
 .ih  y ) ) )
 
Theoremadjval 27541* Value of the adjoint function for 
T in the domain of 
adjh. (Contributed by NM, 19-Feb-2006.) (Revised by Mario Carneiro, 24-Dec-2016.) (New usage is discouraged.)
 |-  ( T  e.  dom  adjh  ->  (
 adjh `  T )  =  ( iota_ u  e.  ( ~H  ^m  ~H ) A. x  e.  ~H  A. y  e.  ~H  ( x  .ih  ( T `  y ) )  =  ( ( u `  x ) 
 .ih  y ) ) )
 
Theoremadjval2 27542* Value of the adjoint function. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  dom  adjh  ->  (
 adjh `  T )  =  ( iota_ u  e.  ( ~H  ^m  ~H ) A. x  e.  ~H  A. y  e.  ~H  ( ( T `
  x )  .ih  y )  =  ( x  .ih  ( u `  y ) ) ) )
 
Theoremcnvadj 27543 The adjoint function equals its converse. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
 |-  `' adjh  =  adjh
 
Theoremfuncnvadj 27544 The converse of the adjoint function is a function. (Contributed by NM, 25-Jan-2006.) (New usage is discouraged.)
 |-  Fun  `'
 adjh
 
Theoremadj1o 27545 The adjoint function maps one-to-one onto its domain. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
 |-  adjh : dom  adjh
 -1-1-onto-> dom  adjh
 
Theoremdmadjrn 27546 The adjoint of an operator belongs to the adjoint function's domain. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  dom  adjh  ->  (
 adjh `  T )  e. 
 dom  adjh )
 
Theoremeigvecval 27547* The set of eigenvectors of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  (
 eigvec `  T )  =  { x  e.  ( ~H  \  0H )  | 
 E. y  e.  CC  ( T `  x )  =  ( y  .h  x ) } )
 
Theoremeigvalfval 27548* The eigenvalues of eigenvectors of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  (
 eigval `  T )  =  ( x  e.  ( eigvec `
  T )  |->  ( ( ( T `  x )  .ih  x ) 
 /  ( ( normh `  x ) ^ 2
 ) ) ) )
 
Theoremspecval 27549* The value of the spectrum of an operator. (Contributed by NM, 11-Apr-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  (
 Lambda `  T )  =  { x  e.  CC  |  -.  ( T  -op  ( x  .op  (  _I  |`  ~H ) ) ) : ~H -1-1-> ~H }
 )
 
Theoremspeccl 27550 The spectrum of an operator is a set of complex numbers. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  (
 Lambda `  T )  C_  CC )
 
Theoremhhlnoi 27551 The linear operators of Hilbert space. (Contributed by NM, 19-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  L  =  ( U 
 LnOp  U )   =>    |- 
 LinOp  =  L
 
Theoremhhnmoi 27552 The norm of an operator in Hilbert space. (Contributed by NM, 19-Nov-2007.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  N  =  ( U
 normOpOLD U )   =>    |-  normop  =  N
 
Theoremhhbloi 27553 A bounded linear operator in Hilbert space. (Contributed by NM, 19-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  B  =  ( U 
 BLnOp  U )   =>    |-  BndLinOp 
 =  B
 
Theoremhh0oi 27554 The zero operator in Hilbert space. (Contributed by NM, 7-Dec-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  Z  =  ( U 
 0op  U )   =>    |- 
 0hop  =  Z
 
Theoremhhcno 27555 The continuous operators of Hilbert space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  D  =  ( normh  o.  -h  )   &    |-  J  =  ( MetOpen `  D )   =>    |-  ConOp  =  ( J  Cn  J )
 
Theoremhhcnf 27556 The continuous functionals of Hilbert space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  D  =  ( normh  o.  -h  )   &    |-  J  =  ( MetOpen `  D )   &    |-  K  =  ( TopOpen ` fld )   =>    |- 
 ConFn  =  ( J  Cn  K )
 
Theoremdmadjrnb 27557 The adjoint of an operator belongs to the adjoint function's domain. (Note: the converse is dependent on our definition of function value, since it uses ndmfv 5905.) (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  dom  adjh  <->  ( adjh `  T )  e.  dom  adjh )
 
Theoremnmoplb 27558 A lower bound for an operator norm. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( T : ~H --> ~H  /\  A  e.  ~H  /\  ( normh `  A )  <_  1 )  ->  ( normh `  ( T `  A ) )  <_  ( normop `  T )
 )
 
Theoremnmopub 27559* An upper bound for an operator norm. (Contributed by NM, 7-Mar-2006.) (New usage is discouraged.)
 |-  (
 ( T : ~H --> ~H  /\  A  e.  RR* )  ->  ( ( normop `  T )  <_  A  <->  A. x  e.  ~H  ( ( normh `  x )  <_  1  ->  ( normh `  ( T `  x ) )  <_  A ) ) )
 
Theoremnmopub2tALT 27560* An upper bound for an operator norm. (Contributed by NM, 12-Apr-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  (
 ( T : ~H --> ~H  /\  ( A  e.  RR  /\  0  <_  A )  /\  A. x  e. 
 ~H  ( normh `  ( T `  x ) ) 
 <_  ( A  x.  ( normh `  x ) ) )  ->  ( normop `  T )  <_  A )
 
Theoremnmopub2tHIL 27561* An upper bound for an operator norm. (Contributed by NM, 13-Dec-2007.) (New usage is discouraged.)
 |-  (
 ( T : ~H --> ~H  /\  ( A  e.  RR  /\  0  <_  A )  /\  A. x  e. 
 ~H  ( normh `  ( T `  x ) ) 
 <_  ( A  x.  ( normh `  x ) ) )  ->  ( normop `  T )  <_  A )
 
Theoremnmopge0 27562 The norm of any Hilbert space operator is nonnegative. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  0 
 <_  ( normop `  T )
 )
 
Theoremnmopgt0 27563 A linear Hilbert space operator that is not identically zero has a positive norm. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  ( ( normop `  T )  =/=  0  <->  0  <  ( normop `  T ) ) )
 
Theoremcnopc 27564* Basic continuity property of a continuous Hilbert space operator. (Contributed by NM, 2-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( T  e.  ConOp  /\  A  e.  ~H  /\  B  e.  RR+ )  ->  E. x  e.  RR+  A. y  e.  ~H  ( ( normh `  ( y  -h  A ) )  <  x  ->  ( normh `  ( ( T `  y )  -h  ( T `  A ) ) )  <  B ) )
 
Theoremlnopl 27565 Basic linearity property of a linear Hilbert space operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
 |-  (
 ( ( T  e.  LinOp  /\  A  e.  CC )  /\  ( B  e.  ~H  /\  C  e.  ~H )
 )  ->  ( T `  ( ( A  .h  B )  +h  C ) )  =  ( ( A  .h  ( T `
  B ) )  +h  ( T `  C ) ) )
 
Theoremunop 27566 Basic inner product property of a unitary operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  UniOp  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( ( T `  A )  .ih  ( T `
  B ) )  =  ( A  .ih  B ) )
 
Theoremunopf1o 27567 A unitary operator in Hilbert space is one-to-one and onto. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
 |-  ( T  e.  UniOp  ->  T : ~H
 -1-1-onto-> ~H )
 
Theoremunopnorm 27568 A unitary operator is idempotent in the norm. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  UniOp  /\  A  e.  ~H )  ->  ( normh `  ( T `  A ) )  =  ( normh `  A )
 )
 
Theoremcnvunop 27569 The inverse (converse) of a unitary operator in Hilbert space is unitary. Theorem in [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
 |-  ( T  e.  UniOp  ->  `' T  e.  UniOp )
 
Theoremunopadj 27570 The inverse (converse) of a unitary operator is its adjoint. Equation 2 of [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  UniOp  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( ( T `  A )  .ih  B )  =  ( A  .ih  ( `' T `  B ) ) )
 
Theoremunoplin 27571 A unitary operator is linear. Theorem in [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
 |-  ( T  e.  UniOp  ->  T  e.  LinOp )
 
Theoremcounop 27572 The composition of two unitary operators is unitary. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
 |-  (
 ( S  e.  UniOp  /\  T  e.  UniOp )  ->  ( S  o.  T )  e.  UniOp )
 
Theoremhmop 27573 Basic inner product property of a Hermitian operator. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  HrmOp  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  ( T `  B ) )  =  ( ( T `
  A )  .ih  B ) )
 
Theoremhmopre 27574 The inner product of the value and argument of a Hermitian operator is real. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  HrmOp  /\  A  e.  ~H )  ->  ( ( T `  A )  .ih  A )  e.  RR )
 
Theoremnmfnlb 27575 A lower bound for a functional norm. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( T : ~H --> CC  /\  A  e.  ~H  /\  ( normh `  A )  <_  1 )  ->  ( abs `  ( T `  A ) )  <_  ( normfn `  T )
 )
 
Theoremnmfnleub 27576* An upper bound for the norm of a functional. (Contributed by NM, 24-May-2006.) (Revised by Mario Carneiro, 7-Sep-2014.) (New usage is discouraged.)
 |-  (
 ( T : ~H --> CC  /\  A  e.  RR* )  ->  ( ( normfn `  T )  <_  A  <->  A. x  e.  ~H  ( ( normh `  x )  <_  1  ->  ( abs `  ( T `  x ) )  <_  A ) ) )
 
Theoremnmfnleub2 27577* An upper bound for the norm of a functional. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
 |-  (
 ( T : ~H --> CC  /\  ( A  e.  RR  /\  0  <_  A )  /\  A. x  e. 
 ~H  ( abs `  ( T `  x ) ) 
 <_  ( A  x.  ( normh `  x ) ) )  ->  ( normfn `  T )  <_  A )
 
Theoremnmfnge0 27578 The norm of any Hilbert space functional is nonnegative. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> CC  ->  0 
 <_  ( normfn `  T )
 )
 
Theoremelnlfn 27579 Membership in the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
 |-  ( T : ~H --> CC  ->  ( A  e.  ( null `  T )  <->  ( A  e.  ~H 
 /\  ( T `  A )  =  0
 ) ) )
 
Theoremelnlfn2 27580 Membership in the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( T : ~H --> CC  /\  A  e.  ( null `  T ) ) 
 ->  ( T `  A )  =  0 )
 
Theoremcnfnc 27581* Basic continuity property of a continuous functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( T  e.  ConFn  /\  A  e.  ~H  /\  B  e.  RR+ )  ->  E. x  e.  RR+  A. y  e.  ~H  ( ( normh `  ( y  -h  A ) )  <  x  ->  ( abs `  ( ( T `  y )  -  ( T `  A ) ) )  <  B ) )
 
Theoremlnfnl 27582 Basic linearity property of a linear functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( ( T  e.  LinFn  /\  A  e.  CC )  /\  ( B  e.  ~H  /\  C  e.  ~H )
 )  ->  ( T `  ( ( A  .h  B )  +h  C ) )  =  ( ( A  x.  ( T `
  B ) )  +  ( T `  C ) ) )
 
Theoremadjcl 27583 Closure of the adjoint of a Hilbert space operator. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  dom  adjh  /\  A  e.  ~H )  ->  ( ( adjh `  T ) `  A )  e. 
 ~H )
 
Theoremadj1 27584 Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  dom  adjh  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  ( T `  B ) )  =  ( ( (
 adjh `  T ) `  A )  .ih  B ) )
 
Theoremadj2 27585 Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  dom  adjh  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( ( T `  A )  .ih  B )  =  ( A  .ih  ( ( adjh `  T ) `  B ) ) )
 
Theoremadjeq 27586* A property that determines the adjoint of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( T : ~H --> ~H  /\  S : ~H --> ~H  /\  A. x  e. 
 ~H  A. y  e.  ~H  ( ( T `  x )  .ih  y )  =  ( x  .ih  ( S `  y ) ) )  ->  ( adjh `  T )  =  S )
 
Theoremadjadj 27587 Double adjoint. Theorem 3.11(iv) of [Beran] p. 106. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  dom  adjh  ->  (
 adjh `  ( adjh `  T ) )  =  T )
 
Theoremadjvalval 27588* Value of the value of the adjoint function. (Contributed by NM, 22-Feb-2006.) (Proof shortened by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
 |-  (
 ( T  e.  dom  adjh  /\  A  e.  ~H )  ->  ( ( adjh `  T ) `  A )  =  ( iota_ w  e.  ~H  A. x  e.  ~H  (
 ( T `  x )  .ih  A )  =  ( x  .ih  w ) ) )
 
Theoremunopadj2 27589 The adjoint of a unitary operator is its inverse (converse). Equation 2 of [AkhiezerGlazman] p. 72. (Contributed by NM, 23-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  UniOp  ->  ( adjh `  T )  =  `' T )
 
Theoremhmopadj 27590 A Hermitian operator is self-adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
 |-  ( T  e.  HrmOp  ->  ( adjh `  T )  =  T )
 
Theoremhmdmadj 27591 Every Hermitian operator has an adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
 |-  ( T  e.  HrmOp  ->  T  e.  dom  adjh )
 
Theoremhmopadj2 27592 An operator is Hermitian iff it is self-adjoint. Definition of Hermitian in [Halmos] p. 41. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)
 |-  ( T  e.  dom  adjh  ->  ( T  e.  HrmOp  <->  ( adjh `  T )  =  T )
 )
 
Theoremhmoplin 27593 A Hermitian operator is linear. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
 |-  ( T  e.  HrmOp  ->  T  e.  LinOp )
 
Theorembrafval 27594* The bra of a vector, expressed as 
<. A  | in Dirac notation. See df-bra 27501. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( bra `  A )  =  ( x  e.  ~H  |->  ( x  .ih  A ) ) )
 
Theorembraval 27595 A bra-ket juxtaposition, expressed as  <. A  |  B >. in Dirac notation, equals the inner product of the vectors. Based on definition of bra in [Prugovecki] p. 186. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( bra `  A ) `  B )  =  ( B  .ih  A ) )
 
Theorembraadd 27596 Linearity property of bra for addition. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( bra `  A ) `  ( B  +h  C ) )  =  ( ( ( bra `  A ) `  B )  +  ( ( bra `  A ) `  C ) ) )
 
Theorembramul 27597 Linearity property of bra for multiplication. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( ( bra `  A ) `  ( B  .h  C ) )  =  ( B  x.  (
 ( bra `  A ) `  C ) ) )
 
Theorembrafn 27598 The bra function is a functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( bra `  A ) : ~H --> CC )
 
Theorembralnfn 27599 The Dirac bra function is a linear functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( bra `  A )  e.  LinFn )
 
Theorembracl 27600 Closure of the bra function. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( bra `  A ) `  B )  e. 
 CC )
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