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Theorem List for Metamath Proof Explorer - 22401-22500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhoaddsubi 22401 Law for sum and difference of Hilbert space operators. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
 |-  R : ~H --> ~H   &    |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( ( R 
 +op  S )  -op  T )  =  ( ( R  -op  T )  +op  S )
 
Theoremhosd1i 22402 Hilbert space operator sum expressed in terms of difference. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
 |-  T : ~H --> ~H   &    |-  U : ~H --> ~H   =>    |-  ( T  +op  U )  =  ( T  -op  ( 0hop  -op  U ) )
 
Theoremhosd2i 22403 Hilbert space operator sum expressed in terms of difference. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
 |-  T : ~H --> ~H   &    |-  U : ~H --> ~H   =>    |-  ( T  +op  U )  =  ( T  -op  ( ( U  -op  U )  -op  U ) )
 
Theoremhopncani 22404 Hilbert space operator cancellation law. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  T : ~H --> ~H   &    |-  U : ~H --> ~H   =>    |-  ( ( T  +op  U )  -op  U )  =  T
 
Theoremhonpcani 22405 Hilbert space operator cancellation law. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
 |-  T : ~H --> ~H   &    |-  U : ~H --> ~H   =>    |-  ( ( T  -op  U )  +op  U )  =  T
 
Theoremhosubeq0i 22406 If the difference between two operators is zero, they are equal. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.)
 |-  T : ~H --> ~H   &    |-  U : ~H --> ~H   =>    |-  ( ( T  -op  U )  =  0hop  <->  T  =  U )
 
Theoremhonpncani 22407 Hilbert space operator cancellation law. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
 |-  R : ~H --> ~H   &    |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( ( R 
 -op  S )  +op  ( S  -op  T ) )  =  ( R  -op  T )
 
Theoremho01i 22408* A condition implying that a Hilbert space operator is identically zero. Lemma 3.2(S8) of [Beran] p. 95. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
 |-  T : ~H --> ~H   =>    |-  ( A. x  e. 
 ~H  A. y  e.  ~H  ( ( T `  x )  .ih  y )  =  0  <->  T  =  0hop )
 
Theoremho02i 22409* A condition implying that a Hilbert space operator is identically zero. Lemma 3.2(S10) of [Beran] p. 95. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
 |-  T : ~H --> ~H   =>    |-  ( A. x  e. 
 ~H  A. y  e.  ~H  ( x  .ih  ( T `
  y ) )  =  0  <->  T  =  0hop )
 
Theoremhoeq1 22410* A condition implying that two Hilbert space operators are equal. Lemma 3.2(S9) of [Beran] p. 95. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( A. x  e.  ~H  A. y  e.  ~H  ( ( S `
  x )  .ih  y )  =  (
 ( T `  x )  .ih  y )  <->  S  =  T ) )
 
Theoremhoeq2 22411* A condition implying that two Hilbert space operators are equal. Lemma 3.2(S11) of [Beran] p. 95. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( A. x  e.  ~H  A. y  e.  ~H  ( x  .ih  ( S `  y ) )  =  ( x 
 .ih  ( T `  y ) )  <->  S  =  T ) )
 
Theoremadjmo 22412* Every Hilbert space operator has at most one adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
 |-  E* u ( u : ~H
 --> ~H  /\  A. x  e.  ~H  A. y  e. 
 ~H  ( x  .ih  ( T `  y ) )  =  ( ( u `  x ) 
 .ih  y ) )
 
Theoremadjsym 22413* Symmetry property of an adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( A. x  e.  ~H  A. y  e.  ~H  ( x  .ih  ( S `  y ) )  =  ( ( T `  x ) 
 .ih  y )  <->  A. x  e.  ~H  A. y  e.  ~H  ( x  .ih  ( T `  y ) )  =  ( ( S `  x )  .ih  y ) ) )
 
Theoremeigrei 22414 A necessary and sufficient condition (that holds when  T is a Hermitian operator) for an eigenvalue  B to be real. Generalization of Equation 1.30 of [Hughes] p. 49. (Contributed by NM, 21-Jan-2005.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  CC   =>    |-  (
 ( ( T `  A )  =  ( B  .h  A )  /\  A  =/=  0h )  ->  ( ( A  .ih  ( T `  A ) )  =  ( ( T `  A ) 
 .ih  A )  <->  B  e.  RR ) )
 
Theoremeigre 22415 A necessary and sufficient condition (that holds when  T is a Hermitian operator) for an eigenvalue  B to be real. Generalization of Equation 1.30 of [Hughes] p. 49. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  ~H 
 /\  B  e.  CC )  /\  ( ( T `
  A )  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  ( ( A  .ih  ( T `  A ) )  =  ( ( T `  A )  .ih  A )  <->  B  e.  RR )
 )
 
Theoremeigposi 22416 A sufficient condition (first conjunct pair, that holds when  T is a positive operator) for an eigenvalue  B (second conjunct pair) to be nonnegative. Remark (ii) in [Hughes] p. 137. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  CC   =>    |-  (
 ( ( ( A 
 .ih  ( T `  A ) )  e. 
 RR  /\  0  <_  ( A  .ih  ( T `  A ) ) ) 
 /\  ( ( T `
  A )  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  ( B  e.  RR  /\  0  <_  B ) )
 
Theoremeigorthi 22417 A necessary and sufficient condition (that holds when  T is a Hermitian operator) for two eigenvectors  A and 
B to be orthogonal. Generalization of Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Jan-2005.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  C  e.  CC   &    |-  D  e.  CC   =>    |-  (
 ( ( ( T `
  A )  =  ( C  .h  A )  /\  ( T `  B )  =  ( D  .h  B ) ) 
 /\  C  =/=  ( * `  D ) ) 
 ->  ( ( A  .ih  ( T `  B ) )  =  ( ( T `  A ) 
 .ih  B )  <->  ( A  .ih  B )  =  0 ) )
 
Theoremeigorth 22418 A necessary and sufficient condition (that holds when  T is a Hermitian operator) for two eigenvectors 
A and  B to be orthogonal. Generalization of Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Mar-2006.) (New usage is discouraged.)
 |-  (
 ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  CC  /\  D  e.  CC ) )  /\  ( ( ( T `
  A )  =  ( C  .h  A )  /\  ( T `  B )  =  ( D  .h  B ) ) 
 /\  C  =/=  ( * `  D ) ) )  ->  ( ( A  .ih  ( T `  B ) )  =  ( ( T `  A )  .ih  B )  <-> 
 ( A  .ih  B )  =  0 )
 )
 
17.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms
 
Definitiondf-nmop 22419* Define the norm of a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
 |-  normop  =  ( t  e.  ( ~H 
 ^m  ~H )  |->  sup ( { x  |  E. z  e.  ~H  ( ( normh `  z )  <_  1  /\  x  =  ( normh `  ( t `  z ) ) ) } ,  RR* ,  <  ) )
 
Definitiondf-cnop 22420* Define the set of continuous operators on Hilbert space. For every "epsilon" ( y) there is an "delta" ( z) such that... (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
 |-  ConOp  =  {
 t  e.  ( ~H 
 ^m  ~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 ) }
 
Definitiondf-lnop 22421* Define the set of linear operators on Hilbert space. (See df-hosum 22310 for definition of operator.) (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
 |-  LinOp  =  {
 t  e.  ( ~H 
 ^m  ~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 ) ) }
 
Definitiondf-bdop 22422 Define the set of bounded linear Hilbert space operators. (See df-hosum 22310 for definition of operator.) (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
 |-  BndLinOp  =  {
 t  e.  LinOp  |  ( normop `
  t )  <  +oo }
 
Definitiondf-unop 22423* Define the set of unitary operators on Hilbert space. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
 |-  UniOp  =  {
 t  |  ( t : ~H -onto-> ~H  /\  A. x  e.  ~H  A. y  e.  ~H  (
 ( t `  x )  .ih  ( t `  y ) )  =  ( x  .ih  y
 ) ) }
 
Definitiondf-hmop 22424* Define the set of Hermitian operators on Hilbert space. Some books call these "symmetric operators" and others call them "self-adjoint operators," sometimes with slightly different technical meanings. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
 |-  HrmOp  =  {
 t  e.  ( ~H 
 ^m  ~H )  |  A. x  e.  ~H  A. y  e.  ~H  ( x  .ih  ( t `  y
 ) )  =  ( ( t `  x )  .ih  y ) }
 
17.6.5  Linear and continuous functionals and norms
 
Definitiondf-nmfn 22425* Define the norm of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
 |-  normfn  =  ( t  e.  ( CC 
 ^m  ~H )  |->  sup ( { x  |  E. z  e.  ~H  ( ( normh `  z )  <_  1  /\  x  =  ( abs `  ( t `  z ) ) ) } ,  RR* ,  <  ) )
 
Definitiondf-nlfn 22426 Define the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
 |-  null  =  ( t  e.  ( CC  ^m  ~H )  |->  ( `' t " { 0 } ) )
 
Definitiondf-cnfn 22427* Define the set of continuous functionals on Hilbert space. For every "epsilon" ( y) there is an "delta" ( z) such that... (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
 |-  ConFn  =  {
 t  e.  ( CC 
 ^m  ~H )  |  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 ) }
 
Definitiondf-lnfn 22428* Define the set of linear functionals on Hilbert space. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
 |-  LinFn  =  {
 t  e.  ( CC 
 ^m  ~H )  |  A. x  e.  CC  A. y  e.  ~H  A. z  e. 
 ~H  ( t `  ( ( x  .h  y )  +h  z
 ) )  =  ( ( x  x.  (
 t `  y )
 )  +  ( t `
  z ) ) }
 
17.6.6  Adjoint
 
Definitiondf-adjh 22429* Define the adjoint of a Hilbert space operator (if it exists). The domain of  adjh is the set of all adjoint operators. Definition of adjoint in [Kalmbach2] p. 8. Unlike Kalmbach (and most authors), we do not demand that the operator be linear, but instead show (in adjbdln 22663) that the adjoint exists for a bounded linear 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  ( ( t `
  x )  .ih  y )  =  ( x  .ih  ( u `  y ) ) ) }
 
17.6.7  Dirac bra-ket notation
 
Definitiondf-bra 22430* 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 22524. The reversal of the inner product arguments not only makes the bra-ket behavior consistent with physics literature (see comments under ax-his3 21663) but is also required in order for the associative law kbass2 22697 to work.

Our definition of bra and the associated outer product df-kb 22431 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 22431* 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 22430, 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 ) ) )
 
17.6.8  Positive operators
 
Definitiondf-leop 22432* 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 ) ) }
 
17.6.9  Eigenvectors, eigenvalues, spectrum
 
Definitiondf-eigvec 22433* Define the eigenvector function. Theorem eleigveccl 22539 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 22434* Define the eigenvalue function. The range of  eigval `  T is the set of eigenvalues of Hilbert space operator  T. Theorem eigvalcl 22541 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 22435* 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
 } )
 
17.6.10  Theorems about operators and functionals
 
Theoremnmopval 22436* 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 22437* 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 22438* 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 22439 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 22440 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 22441 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 22442 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 22443* The set in the supremum of the operator norm definition df-nmop 22419 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 22444* The set in the supremum of the operator norm definition df-nmop 22419 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 22445* The set in the supremum of the operator norm definition df-nmop 22419 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 22446 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 22447 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 22448 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 22449 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 22450 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 22451 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 22452* 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 22453* 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 22454 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 22455 The class of Hermitian operators is a set. (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
 |-  HrmOp  e.  _V
 
Theoremnmfnval 22456* 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 22457* The set in the supremum of the functional norm definition df-nmfn 22425 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 22458* The set in the supremum of the functional norm definition df-nmfn 22425 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 22459 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 22460 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 22461 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 22462* 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 22463* 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 22464 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 22465* 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 22466 Functionality of the adjoint function. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
 |-  Fun  adjh
 
Theoremdmadjss 22467 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 22468 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 22469* 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 22470* 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 22471* 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 22472 The adjoint function equals its converse. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
 |-  `' adjh  =  adjh
 
Theoremfuncnvadj 22473 The converse of the adjoint function is a function. (Contributed by NM, 25-Jan-2006.) (New usage is discouraged.)
 |-  Fun  `'
 adjh
 
Theoremadj1o 22474 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 22475 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 22476* 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 22477* 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 22478* 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 22479 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 22480 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 22481 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 22482 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 22483 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 22484 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 22485 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 22486 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 5552.) (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  dom  adjh  <->  ( adjh `  T )  e.  dom  adjh )
 
Theoremnmoplb 22487 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 22488* 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 22489* 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 22490* 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 22491 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 22492 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 22493* 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 22494 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 22495 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 22496 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 22497 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 22498 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 22499 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 22500 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 )
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