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Theorem List for Metamath Proof Explorer - 22301-22400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmayetes3i 22301 Mayet's equation E^*3, derived from E3. Solution, for n = 3, to open problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. (Contributed by NM, 10-May-2009.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   &    |-  F  e.  CH   &    |-  G  e.  CH   &    |-  R  e.  CH   &    |-  A  C_  ( _|_ `  C )   &    |-  A  C_  ( _|_ `  F )   &    |-  C  C_  ( _|_ `  F )   &    |-  A  C_  ( _|_ `  B )   &    |-  C  C_  ( _|_ `  D )   &    |-  F  C_  ( _|_ `  G )   &    |-  R  C_  ( _|_ `  X )   &    |-  X  =  ( ( A  vH  C )  vH  F )   &    |-  Y  =  ( (
 ( A  vH  B )  i^i  ( C  vH  D ) )  i^i  ( F  vH  G ) )   &    |-  Z  =  ( ( B  vH  D )  vH  G )   =>    |-  ( ( X 
 vH  R )  i^i 
 Y )  C_  ( Z  vH  R )
 
17.6  Operators on Hilbert spaces
 
17.6.1  Operator sum, difference, and scalar multiplication

Note on operators. Unlike some authors, we use the term "operator" to mean any function from  ~H to  ~H. This is the definition of operator in [Hughes] p. 14, the definition of operator in [AkhiezerGlazman] p. 30, and the definition of operator in [Goldberg] p. 10. For Reed and Simon, an operator is linear (definition of operator in [ReedSimon] p. 2). For Halmos, an operator is bounded and linear (definition of operator in [Halmos] p. 35). For Kalmbach and Beran, an operator is continuous and linear (definition of operator in [Kalmbach] p. 353; definition of operator in [Beran] p. 99). Note that "bounded and linear" and "continuous and linear" are equivalent by lncnbd 22610.

 
Definitiondf-hosum 22302* Define the sum of two Hilbert space operators. Definition of [Beran] p. 111. (Contributed by NM, 9-Nov-2000.) (New usage is discouraged.)
 |-  +op  =  ( f  e.  ( ~H  ^m  ~H ) ,  g  e.  ( ~H 
 ^m  ~H )  |->  ( x  e.  ~H  |->  ( ( f `  x )  +h  ( g `  x ) ) ) )
 
Definitiondf-homul 22303* Define the scalar product with a Hilbert space operator. Definition of [Beran] p. 111. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
 |-  .op  =  ( f  e.  CC ,  g  e.  ( ~H  ^m  ~H )  |->  ( x  e.  ~H  |->  ( f  .h  ( g `
  x ) ) ) )
 
Definitiondf-hodif 22304* Define the difference of two Hilbert space operators. Definition of [Beran] p. 111. (Contributed by NM, 9-Nov-2000.) (New usage is discouraged.)
 |-  -op  =  ( f  e.  ( ~H  ^m  ~H ) ,  g  e.  ( ~H 
 ^m  ~H )  |->  ( x  e.  ~H  |->  ( ( f `  x )  -h  ( g `  x ) ) ) )
 
Definitiondf-hfsum 22305* Define the sum of two Hilbert space functionals. Definition of [Beran] p. 111. Note that unlike some authors, we define a functional as any function from  ~H to  CC, not just linear (or bounded linear) ones. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
 |-  +fn  =  ( f  e.  ( CC  ^m  ~H ) ,  g  e.  ( CC 
 ^m  ~H )  |->  ( x  e.  ~H  |->  ( ( f `  x )  +  ( g `  x ) ) ) )
 
Definitiondf-hfmul 22306* Define the scalar product with a Hilbert space functional. Definition of [Beran] p. 111. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
 |-  .fn  =  ( f  e.  CC ,  g  e.  ( CC  ^m  ~H )  |->  ( x  e.  ~H  |->  ( f  x.  ( g `
  x ) ) ) )
 
Theoremhosmval 22307* Value of the sum of two Hilbert space operators. (Contributed by NM, 9-Nov-2000.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S 
 +op  T )  =  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) ) )
 
Theoremhommval 22308* Value of the scalar product with a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T )  =  ( x  e.  ~H  |->  ( A  .h  ( T `  x ) ) ) )
 
Theoremhodmval 22309* Value of the difference of two Hilbert space operators. (Contributed by NM, 9-Nov-2000.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S 
 -op  T )  =  ( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x ) ) ) )
 
Theoremhfsmval 22310* Value of the sum of two Hilbert space functionals. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> CC  /\  T : ~H --> CC )  ->  ( S 
 +fn  T )  =  ( x  e.  ~H  |->  ( ( S `  x )  +  ( T `  x ) ) ) )
 
Theoremhfmmval 22311* Value of the scalar product with a Hilbert space functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T : ~H --> CC )  ->  ( A  .fn  T )  =  ( x  e.  ~H  |->  ( A  x.  ( T `  x ) ) ) )
 
Theoremhosval 22312 Value of the sum of two Hilbert space operators. (Contributed by NM, 10-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  A  e.  ~H )  ->  ( ( S 
 +op  T ) `  A )  =  ( ( S `  A )  +h  ( T `  A ) ) )
 
Theoremhomval 22313 Value of the scalar product with a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T : ~H --> ~H  /\  B  e.  ~H )  ->  ( ( A  .op  T ) `  B )  =  ( A  .h  ( T `  B ) ) )
 
Theoremhodval 22314 Value of the difference of two Hilbert space operators. (Contributed by NM, 10-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  A  e.  ~H )  ->  ( ( S 
 -op  T ) `  A )  =  ( ( S `  A )  -h  ( T `  A ) ) )
 
Theoremhfsval 22315 Value of the sum of two Hilbert space functionals. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> CC  /\  T : ~H --> CC  /\  A  e.  ~H )  ->  ( ( S 
 +fn  T ) `  A )  =  ( ( S `  A )  +  ( T `  A ) ) )
 
Theoremhfmval 22316 Value of the scalar product with a Hilbert space functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T : ~H --> CC  /\  B  e.  ~H )  ->  ( ( A  .fn  T ) `  B )  =  ( A  x.  ( T `  B ) ) )
 
Theoremhoscl 22317 Closure of the sum of two Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
 |-  (
 ( ( S : ~H
 --> ~H  /\  T : ~H
 --> ~H )  /\  A  e.  ~H )  ->  (
 ( S  +op  T ) `  A )  e. 
 ~H )
 
Theoremhomcl 22318 Closure of the scalar product of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T : ~H --> ~H  /\  B  e.  ~H )  ->  ( ( A  .op  T ) `  B )  e.  ~H )
 
Theoremhodcl 22319 Closure of the difference of two Hilbert space operators. (Contributed by NM, 15-Nov-2002.) (New usage is discouraged.)
 |-  (
 ( ( S : ~H
 --> ~H  /\  T : ~H
 --> ~H )  /\  A  e.  ~H )  ->  (
 ( S  -op  T ) `  A )  e. 
 ~H )
 
17.6.2  Zero and identity operators
 
Definitiondf-h0op 22320 Define the Hilbert space zero operator. See df0op2 22324 for alternate definition. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
 |-  0hop  =  ( proj  h `  0H )
 
Definitiondf-iop 22321 Define the Hilbert space identity operator. See dfiop2 22325 for alternate definition. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
 |-  Iop  =  ( proj  h `  ~H )
 
Theoremho0val 22322 Value of the zero Hilbert space operator (null projector). Remark in [Beran] p. 111. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  (
 0hop `  A )  =  0h )
 
Theoremho0f 22323 Functionality of the zero Hilbert space operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  0hop : ~H --> ~H
 
Theoremdf0op2 22324 Alternate definition of Hilbert space zero operator. (Contributed by NM, 7-Aug-2006.) (New usage is discouraged.)
 |-  0hop  =  ( ~H  X.  0H )
 
Theoremdfiop2 22325 Alternate definition of Hilbert space identity operator. (Contributed by NM, 7-Aug-2006.) (New usage is discouraged.)
 |-  Iop  =  (  _I  |`  ~H )
 
Theoremhoif 22326 Functionality of the Hilbert space identity operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)
 |-  Iop  : ~H
 -1-1-onto-> ~H
 
Theoremhoival 22327 The value of the Hilbert space identity operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( 
 Iop  `  A )  =  A )
 
Theoremhoico1 22328 Composition with the Hilbert space identity operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  ( T  o.  Iop  )  =  T )
 
Theoremhoico2 22329 Composition with the Hilbert space identity operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  ( 
 Iop  o.  T )  =  T )
 
17.6.3  Operations on Hilbert space operators
 
Theoremhoaddcl 22330 The sum of Hilbert space operators is an operator. (Contributed by NM, 21-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S 
 +op  T ) : ~H --> ~H )
 
Theoremhomulcl 22331 The scalar product of a Hilbert space operator is an operator. (Contributed by NM, 21-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T ) : ~H --> ~H )
 
Theoremhoeq 22332* Equality of Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A. x  e.  ~H  ( T `  x )  =  ( U `  x ) 
 <->  T  =  U ) )
 
Theoremhoeqi 22333* Equality of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( A. x  e. 
 ~H  ( S `  x )  =  ( T `  x )  <->  S  =  T )
 
Theoremhoscli 22334 Closure of Hilbert space operator sum. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( A  e.  ~H  ->  ( ( S  +op  T ) `  A )  e.  ~H )
 
Theoremhodcli 22335 Closure of Hilbert space operator difference. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( A  e.  ~H  ->  ( ( S  -op  T ) `  A )  e.  ~H )
 
Theoremhocoi 22336 Composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( A  e.  ~H  ->  ( ( S  o.  T ) `  A )  =  ( S `  ( T `  A ) ) )
 
Theoremhococli 22337 Closure of composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( A  e.  ~H  ->  ( ( S  o.  T ) `  A )  e.  ~H )
 
Theoremhocofi 22338 Mapping of composition of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( S  o.  T ) : ~H --> ~H
 
Theoremhocofni 22339 Functionality of composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( S  o.  T )  Fn  ~H
 
Theoremhoaddcli 22340 Mapping of sum of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( S  +op  T ) : ~H --> ~H
 
Theoremhosubcli 22341 Mapping of difference of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( S  -op  T ) : ~H --> ~H
 
Theoremhoaddfni 22342 Functionality of sum of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( S  +op  T )  Fn  ~H
 
Theoremhosubfni 22343 Functionality of difference of Hilbert space operators. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( S  -op  T )  Fn  ~H
 
Theoremhoaddcomi 22344 Commutativity of sum of Hilbert space operators. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
 |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( S  +op  T )  =  ( T  +op  S )
 
Theoremhosubcl 22345 Mapping of difference of Hilbert space operators. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S 
 -op  T ) : ~H --> ~H )
 
Theoremhoaddcom 22346 Commutativity of sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S 
 +op  T )  =  ( T  +op  S )
 )
 
Theoremhodsi 22347 Relationship between Hilbert space operator difference and sum. (Contributed by NM, 17-Nov-2000.) (New usage is discouraged.)
 |-  R : ~H --> ~H   &    |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( ( R 
 -op  S )  =  T  <->  ( S  +op  T )  =  R )
 
Theoremhoaddassi 22348 Associativity of sum of Hilbert space operators. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
 |-  R : ~H --> ~H   &    |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( ( R 
 +op  S )  +op  T )  =  ( R  +op  ( S  +op  T ) )
 
Theoremhoadd12i 22349 Commutative/associative law for Hilbert space operator sum that swaps the first two terms. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
 |-  R : ~H --> ~H   &    |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( R  +op  ( S  +op  T ) )  =  ( S 
 +op  ( R  +op  T ) )
 
Theoremhoadd32i 22350 Commutative/associative law for Hilbert space operator sum that swaps the second and third terms. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.)
 |-  R : ~H --> ~H   &    |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( ( R 
 +op  S )  +op  T )  =  ( ( R  +op  T )  +op  S )
 
Theoremhocadddiri 22351 Distributive law for Hilbert space operator sum. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
 |-  R : ~H --> ~H   &    |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( ( R 
 +op  S )  o.  T )  =  ( ( R  o.  T )  +op  ( S  o.  T ) )
 
Theoremhocsubdiri 22352 Distributive law for Hilbert space operator difference. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
 |-  R : ~H --> ~H   &    |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( ( R 
 -op  S )  o.  T )  =  ( ( R  o.  T )  -op  ( S  o.  T ) )
 
Theoremho2coi 22353 Double composition of Hilbert space operators. (Contributed by NM, 1-Dec-2000.) (New usage is discouraged.)
 |-  R : ~H --> ~H   &    |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( A  e.  ~H 
 ->  ( ( ( R  o.  S )  o.  T ) `  A )  =  ( R `  ( S `  ( T `  A ) ) ) )
 
Theoremhoaddass 22354 Associativity of sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( R : ~H --> ~H  /\  S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( ( R  +op  S )  +op  T )  =  ( R  +op  ( S  +op  T ) ) )
 
Theoremhoadd32 22355 Commutative/associative law for Hilbert space operator sum that swaps the second and third terms. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( R : ~H --> ~H  /\  S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( ( R  +op  S )  +op  T )  =  ( ( R  +op  T )  +op  S ) )
 
Theoremhoadd4 22356 Rearrangement of 4 terms in a sum 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 ) ) )
 
Theoremhocsubdir 22357 Distributive law for Hilbert space operator difference. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( R : ~H --> ~H  /\  S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( ( R  -op  S )  o.  T )  =  ( ( R  o.  T )  -op  ( S  o.  T ) ) )
 
Theoremhoaddid1i 22358 Sum of a Hilbert space operator with the zero operator. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
 |-  T : ~H --> ~H   =>    |-  ( T  +op  0hop )  =  T
 
Theoremhodidi 22359 Difference of a Hilbert space operator from itself. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  T : ~H --> ~H   =>    |-  ( T  -op  T )  =  0hop
 
Theoremho0coi 22360 Composition of the zero operator and a Hilbert space operator. (Contributed by NM, 9-Aug-2006.) (New usage is discouraged.)
 |-  T : ~H --> ~H   =>    |-  ( 0hop  o.  T )  =  0hop
 
Theoremhoid1i 22361 Composition of Hilbert space operator with unit identity. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
 |-  T : ~H --> ~H   =>    |-  ( T  o.  Iop  )  =  T
 
Theoremhoid1ri 22362 Composition of Hilbert space operator with unit identity. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
 |-  T : ~H --> ~H   =>    |-  (  Iop  o.  T )  =  T
 
Theoremhoaddid1 22363 Sum of a Hilbert space operator with the zero operator. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  ( T  +op  0hop )  =  T )
 
Theoremhodid 22364 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 22365 A Hilbert space operator is not empty. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  -.  T  =  (/) )
 
Theoremhodseqi 22366 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 22367 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 22368 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 22369 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 22370 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 22371 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 22372 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 22373 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 22374 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 22375 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 22376 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 22377 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 22378 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 22379 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 22380 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 22381 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 22382 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 22383 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 22384 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 22385 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 22386 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 22387 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 22388 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 22389 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 22390 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 22391 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 22392 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 22393 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 22394 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 22395 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 22396 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 22397 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 22398 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 22399 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 22400* 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 )
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