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Theorem List for Metamath Proof Explorer - 23201-23300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
18.6.7  Dirac bra-ket notation
 
Definitiondf-bra 23201* 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 23295. The reversal of the inner product arguments not only makes the bra-ket behavior consistent with physics literature (see comments under ax-his3 22434) but is also required in order for the associative law kbass2 23468 to work.

Our definition of bra and the associated outer product df-kb 23202 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 23202* 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 23201, 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 ) ) )
 
18.6.8  Positive operators
 
Definitiondf-leop 23203* 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 ) ) }
 
18.6.9  Eigenvectors, eigenvalues, spectrum
 
Definitiondf-eigvec 23204* Define the eigenvector function. Theorem eleigveccl 23310 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 23205* Define the eigenvalue function. The range of  eigval `  T is the set of eigenvalues of Hilbert space operator  T. Theorem eigvalcl 23312 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 23206* 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
 } )
 
18.6.10  Theorems about operators and functionals
 
Theoremnmopval 23207* 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 23208* 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 23209* 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 23210 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 23211 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 23212 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 23213 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 23214* The set in the supremum of the operator norm definition df-nmop 23190 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 )
 
TheoremnmopsetretHIL 23215* The set in the supremum of the operator norm definition df-nmop 23190 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 23216* The set in the supremum of the operator norm definition df-nmop 23190 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 23217 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 23218 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 23219 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 23220 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 23221 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 23222 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 23223* 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 23224* 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 23225 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 23226 The class of Hermitian operators is a set. (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
 |-  HrmOp  e.  _V
 
Theoremnmfnval 23227* 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 23228* The set in the supremum of the functional norm definition df-nmfn 23196 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 23229* The set in the supremum of the functional norm definition df-nmfn 23196 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 23230 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 23231 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 23232 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 23233* 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 23234* 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 23235 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 23236* 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 23237 Functionality of the adjoint function. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
 |-  Fun  adjh
 
Theoremdmadjss 23238 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 23239 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 23240* 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 23241* 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 23242* 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 23243 The adjoint function equals its converse. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
 |-  `' adjh  =  adjh
 
Theoremfuncnvadj 23244 The converse of the adjoint function is a function. (Contributed by NM, 25-Jan-2006.) (New usage is discouraged.)
 |-  Fun  `'
 adjh
 
Theoremadj1o 23245 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 23246 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 23247* 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 23248* 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 23249* 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 23250 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 23251 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 23252 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
 normOp OLD U )   =>    |-  normop  =  N
 
Theoremhhbloi 23253 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 23254 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 23255 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 23256 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 23257 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 5695.) (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  dom  adjh  <->  ( adjh `  T )  e.  dom  adjh )
 
Theoremnmoplb 23258 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 23259* 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 23260* An upper bound for an operator norm. (Contributed by NM, 12-Apr-2006.) (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 )
 
Theoremnmopub2tHIL 23261* 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 23262 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 23263 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 23264* 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 23265 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 23266 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 23267 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 23268 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 23269 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 23270 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 23271 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 23272 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 23273 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 23274 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 23275 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 23276* 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 23277* 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 23278 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 23279 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 23280 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 23281* 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 23282 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 23283 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 23284 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 23285 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 23286* 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 23287 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 23288* 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 23289 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 23290 A Hermitian operator is self-adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
 |-  ( T  e.  HrmOp  ->  ( adjh `  T )  =  T )
 
Theoremhmdmadj 23291 Every Hermitian operator has an adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
 |-  ( T  e.  HrmOp  ->  T  e.  dom  adjh )
 
Theoremhmopadj2 23292 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 23293 A Hermitian operator is linear. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
 |-  ( T  e.  HrmOp  ->  T  e.  LinOp )
 
Theorembrafval 23294* The bra of a vector, expressed as 
<. A  | in Dirac notation. See df-bra 23201. (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 23295 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 23296 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 23297 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 23298 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 23299 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 23300 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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