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Theorem List for Metamath Proof Explorer - 27501-27600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhodid 27501 Difference of a Hilbert space operator from itself. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  ( T  -op  T )  =  0hop )
 
Theoremhon0 27502 A Hilbert space operator is not empty. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  -.  T  =  (/) )
 
Theoremhodseqi 27503 Subtraction and addition of equal Hilbert space operators. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( S  +op  ( T  -op  S ) )  =  T
 
Theoremho0subi 27504 Subtraction of Hilbert space operators expressed in terms of difference from zero. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( S  -op  T )  =  ( S  +op  ( 0hop  -op  T ) )
 
Theoremhonegsubi 27505 Relationship between Hilbert operator addition and subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( S  +op  ( -u 1  .op  T )
 )  =  ( S 
 -op  T )
 
Theoremho0sub 27506 Subtraction of Hilbert space operators expressed in terms of difference from zero. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S 
 -op  T )  =  ( S  +op  ( 0hop  -op 
 T ) ) )
 
Theoremhosubid1 27507 The zero operator subtracted from a Hilbert space operator. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  ( T  -op  0hop )  =  T )
 
Theoremhonegsub 27508 Relationship between Hilbert space operator addition and subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( T 
 +op  ( -u 1  .op  U ) )  =  ( T  -op  U ) )
 
Theoremhomulid2 27509 An operator equals its scalar product with one. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  ( 1  .op  T )  =  T )
 
Theoremhomco1 27510 Associative law for scalar product and composition of operators. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( A  .op  T )  o.  U )  =  ( A  .op  ( T  o.  U ) ) )
 
Theoremhomulass 27511 Scalar product associative law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  ->  ( ( A  x.  B )  .op  T )  =  ( A  .op  ( B  .op  T ) ) )
 
Theoremhoadddi 27512 Scalar product distributive law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A  .op  ( T  +op  U ) )  =  ( ( A 
 .op  T )  +op  ( A  .op  U ) ) )
 
Theoremhoadddir 27513 Scalar product reverse distributive law for Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  ->  ( ( A  +  B )  .op  T )  =  ( ( A 
 .op  T )  +op  ( B  .op  T ) ) )
 
Theoremhomul12 27514 Swap first and second factors in a nested operator scalar product. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  ( B  .op  T ) )  =  ( B  .op  ( A  .op  T ) ) )
 
Theoremhonegneg 27515 Double negative of a Hilbert space operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  (
 -u 1  .op  ( -u 1  .op  T )
 )  =  T )
 
Theoremhosubneg 27516 Relationship between operator subtraction and negative. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( T 
 -op  ( -u 1  .op  U ) )  =  ( T  +op  U ) )
 
Theoremhosubdi 27517 Scalar product distributive law for operator difference. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A  .op  ( T  -op  U ) )  =  ( ( A 
 .op  T )  -op  ( A  .op  U ) ) )
 
Theoremhonegdi 27518 Distribution of negative over addition. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( -u 1  .op  ( T  +op  U ) )  =  ( ( -u 1  .op  T )  +op  ( -u 1  .op  U ) ) )
 
Theoremhonegsubdi 27519 Distribution of negative over subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( -u 1  .op  ( T  -op  U ) )  =  ( ( -u 1  .op  T )  +op  U ) )
 
Theoremhonegsubdi2 27520 Distribution of negative over subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( -u 1  .op  ( T  -op  U ) )  =  ( U  -op  T ) )
 
Theoremhosubsub2 27521 Law for double subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( S 
 -op  ( T  -op  U ) )  =  ( S  +op  ( U  -op  T ) ) )
 
Theoremhosub4 27522 Rearrangement of 4 terms in a mixed addition and subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( ( R : ~H
 --> ~H  /\  S : ~H
 --> ~H )  /\  ( T : ~H --> ~H  /\  U : ~H --> ~H )
 )  ->  ( ( R  +op  S )  -op  ( T  +op  U ) )  =  ( ( R  -op  T ) 
 +op  ( S  -op  U ) ) )
 
Theoremhosubadd4 27523 Rearrangement of 4 terms in a mixed addition and subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( ( R : ~H
 --> ~H  /\  S : ~H
 --> ~H )  /\  ( T : ~H --> ~H  /\  U : ~H --> ~H )
 )  ->  ( ( R  -op  S )  -op  ( T  -op  U ) )  =  ( ( R  +op  U )  -op  ( S  +op  T ) ) )
 
Theoremhoaddsubass 27524 Associative-type law for addition and subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( S  +op  T )  -op  U )  =  ( S  +op  ( T  -op  U ) ) )
 
Theoremhoaddsub 27525 Law for operator addition and subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( S  +op  T )  -op  U )  =  ( ( S  -op  U )  +op  T ) )
 
Theoremhosubsub 27526 Law for double subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( S 
 -op  ( T  -op  U ) )  =  ( ( S  -op  T )  +op  U ) )
 
Theoremhosubsub4 27527 Law for double subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( S  -op  T ) 
 -op  U )  =  ( S  -op  ( T 
 +op  U ) ) )
 
Theoremho2times 27528 Two times a Hilbert space operator. (Contributed by NM, 26-Aug-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  ( 2  .op  T )  =  ( T  +op  T ) )
 
Theoremhoaddsubassi 27529 Associativity of 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  ( S  -op  T ) )
 
Theoremhoaddsubi 27530 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 27531 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 27532 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 27533 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 27534 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 27535 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 27536 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 27537* 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 27538* 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 27539* 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 27540* 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 27541* 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 27542* 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 27543 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 27544 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 27545 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 27546 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 27547 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 )
 )
 
20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms
 
Definitiondf-nmop 27548* 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 27549* Define the set of continuous operators on Hilbert space. For every "epsilon" ( y) there is a "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 27550* Define the set of linear operators on Hilbert space. (See df-hosum 27439 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 27551 Define the set of bounded linear Hilbert space operators. (See df-hosum 27439 for definition of operator.) (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
 |-  BndLinOp  =  {
 t  e.  LinOp  |  ( normop `
  t )  < +oo }
 
Definitiondf-unop 27552* 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 27553* 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 ) }
 
20.6.5  Linear and continuous functionals and norms
 
Definitiondf-nmfn 27554* 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 27555 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 27556* Define the set of continuous functionals on Hilbert space. For every "epsilon" ( y) there is a "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 27557* 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 ) ) }
 
20.6.6  Adjoint
 
Definitiondf-adjh 27558* 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 27792) 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 ) ) ) }
 
20.6.7  Dirac bra-ket notation
 
Definitiondf-bra 27559* 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 27653. The reversal of the inner product arguments not only makes the bra-ket behavior consistent with physics literature (see comments under ax-his3 26793) but is also required in order for the associative law kbass2 27826 to work.

Our definition of bra and the associated outer product df-kb 27560 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 27560* 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 27559, 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 27561* 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 27562* Define the eigenvector function. Theorem eleigveccl 27668 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 27563* Define the eigenvalue function. The range of  eigval `  T is the set of eigenvalues of Hilbert space operator  T. Theorem eigvalcl 27670 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 27564* 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 27565* 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 27566* 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 27567* 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 27568 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 27569 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 27570 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 27571 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 27572* The set in the supremum of the operator norm definition df-nmop 27548 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 27573* The set in the supremum of the operator norm definition df-nmop 27548 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 27574* The set in the supremum of the operator norm definition df-nmop 27548 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 27575 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 27576 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 27577 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 27578 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 27579 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 27580 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 27581* 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 27582* 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 27583 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 27584 The class of Hermitian operators is a set. (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
 |-  HrmOp  e.  _V
 
Theoremnmfnval 27585* 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 27586* The set in the supremum of the functional norm definition df-nmfn 27554 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 27587* The set in the supremum of the functional norm definition df-nmfn 27554 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 27588 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 27589 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 27590 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 27591* 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 27592* 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 27593 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 27594* 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 27595 Functionality of the adjoint function. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
 |-  Fun  adjh
 
Theoremdmadjss 27596 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 27597 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 27598* 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 27599* 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 27600* 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 ) ) ) )
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