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Theorem List for Metamath Proof Explorer - 22501-22600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelnlfn2 22501 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 22502* 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 22503 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 22504 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 22505 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 22506 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 22507* 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 22508 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 22509* 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 22510 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 22511 A Hermitian operator is self-adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
 |-  ( T  e.  HrmOp  ->  ( adjh `  T )  =  T )
 
Theoremhmdmadj 22512 Every Hermitian operator has an adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
 |-  ( T  e.  HrmOp  ->  T  e.  dom  adjh )
 
Theoremhmopadj2 22513 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 22514 A Hermitian operator is linear. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
 |-  ( T  e.  HrmOp  ->  T  e.  LinOp )
 
Theorembrafval 22515* The bra of a vector, expressed as 
<. A  | in Dirac notation. See df-bra 22422. (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 22516 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 22517 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 22518 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 22519 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 22520 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 22521 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 )
 
Theorembra0 22522 The Dirac bra of the zero vector. (Contributed by NM, 25-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
 |-  ( bra `  0h )  =  ( ~H  X.  {
 0 } )
 
Theorembrafnmul 22523 Anti-linearity property of bra functional for multiplication. (Contributed by NM, 31-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  ~H )  ->  ( bra `  ( A  .h  B ) )  =  ( ( * `
  A )  .fn  ( bra `  B )
 ) )
 
Theoremkbfval 22524* The outer product of two vectors, expressed as  |  A >.  <. B  | in Dirac notation. See df-kb 22423. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  ketbra  B )  =  ( x  e. 
 ~H  |->  ( ( x 
 .ih  B )  .h  A ) ) )
 
Theoremkbop 22525 The outer product of two vectors, expressed as  |  A >.  <. B  | in Dirac notation, is an operator. (Contributed by NM, 30-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  ketbra  B ) : ~H --> ~H )
 
Theoremkbval 22526 The value of the operator resulting from the outer product  |  A >.  <. B  | of two vectors. Equation 8.1 of [Prugovecki] p. 376. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  ketbra  B ) `  C )  =  ( ( C 
 .ih  B )  .h  A ) )
 
Theoremkbmul 22527 Multiplication property of outer product. (Contributed by NM, 31-May-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  .h  B )  ketbra  C )  =  ( B  ketbra  ( ( * `  A )  .h  C ) ) )
 
Theoremkbpj 22528 If a vector  A has norm 1, the outer product  |  A >.  <. A  | is the projector onto the subspace spanned by  A. http://en.wikipedia.org/wiki/Bra-ket#Linear%5Foperators. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  ( normh `  A )  =  1 )  ->  ( A  ketbra  A )  =  ( proj  h `  ( span `  { A }
 ) ) )
 
Theoremeleigvec 22529* Membership in 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  ->  ( A  e.  ( eigvec `  T )  <->  ( A  e.  ~H 
 /\  A  =/=  0h  /\ 
 E. x  e.  CC  ( T `  A )  =  ( x  .h  A ) ) ) )
 
Theoremeleigvec2 22530 Membership in the set of eigenvectors of a Hilbert space operator. (Contributed by NM, 18-Mar-2006.) (New usage is discouraged.)
 |-  ( T : ~H --> ~H  ->  ( A  e.  ( eigvec `  T )  <->  ( A  e.  ~H 
 /\  A  =/=  0h  /\  ( T `  A )  e.  ( span ` 
 { A } )
 ) ) )
 
Theoremeleigveccl 22531 Closure of an eigenvector of a Hilbert space operator. (Contributed by NM, 23-Mar-2006.) (New usage is discouraged.)
 |-  (
 ( T : ~H --> ~H  /\  A  e.  ( eigvec `
  T ) ) 
 ->  A  e.  ~H )
 
Theoremeigvalval 22532 The eigenvalue of an eigenvector of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
 |-  (
 ( T : ~H --> ~H  /\  A  e.  ( eigvec `
  T ) ) 
 ->  ( ( eigval `  T ) `  A )  =  ( ( ( T `
  A )  .ih  A )  /  ( (
 normh `  A ) ^
 2 ) ) )
 
Theoremeigvalcl 22533 An eigenvalue is a complex number. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
 |-  (
 ( T : ~H --> ~H  /\  A  e.  ( eigvec `
  T ) ) 
 ->  ( ( eigval `  T ) `  A )  e. 
 CC )
 
Theoremeigvec1 22534 Property of an eigenvector. (Contributed by NM, 12-Mar-2006.) (New usage is discouraged.)
 |-  (
 ( T : ~H --> ~H  /\  A  e.  ( eigvec `
  T ) ) 
 ->  ( ( T `  A )  =  (
 ( ( eigval `  T ) `  A )  .h  A )  /\  A  =/=  0h ) )
 
Theoremeighmre 22535 The eigenvalues of a Hermitian operator are real. Equation 1.30 of [Hughes] p. 49. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  HrmOp  /\  A  e.  ( eigvec `  T ) )  ->  ( ( eigval `  T ) `  A )  e. 
 RR )
 
Theoremeighmorth 22536 Eigenvectors of a Hermitian operator with distinct eigenvalues are orthogonal. Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Mar-2006.) (New usage is discouraged.)
 |-  (
 ( ( T  e.  HrmOp  /\  A  e.  ( eigvec `  T ) )  /\  ( B  e.  ( eigvec `
  T )  /\  ( ( eigval `  T ) `  A )  =/=  ( ( eigval `  T ) `  B ) ) )  ->  ( A  .ih  B )  =  0 )
 
Theoremnmopnegi 22537 Value of the norm of the negative of a Hilbert space operator. Unlike nmophmi 22603, the operator does not have to be bounded. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  T : ~H --> ~H   =>    |-  ( normop `  ( -u 1  .op  T ) )  =  ( normop `  T )
 
Theoremlnop0 22538 The value of a linear Hilbert space operator at zero is zero. Remark in [Beran] p. 99. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)
 |-  ( T  e.  LinOp  ->  ( T `  0h )  =  0h )
 
Theoremlnopmul 22539 Multiplicative property of a linear Hilbert space operator. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  LinOp  /\  A  e.  CC  /\  B  e.  ~H )  ->  ( T `  ( A  .h  B ) )  =  ( A  .h  ( T `  B ) ) )
 
Theoremlnopli 22540 Basic scalar product property of a linear Hilbert space operator. (Contributed by NM, 23-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 ) ) )
 
Theoremlnopfi 22541 A linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 23-Jan-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   =>    |-  T : ~H --> ~H
 
Theoremlnop0i 22542 The value of a linear Hilbert space operator at zero is zero. Remark in [Beran] p. 99. (Contributed by NM, 11-May-2005.) (New usage is discouraged.)
 |-  T  e.  LinOp   =>    |-  ( T `  0h )  =  0h
 
Theoremlnopaddi 22543 Additive property of a linear Hilbert space operator. (Contributed by NM, 11-May-2005.) (New usage is discouraged.)
 |-  T  e.  LinOp   =>    |-  ( ( A  e.  ~H 
 /\  B  e.  ~H )  ->  ( T `  ( A  +h  B ) )  =  ( ( T `  A )  +h  ( T `  B ) ) )
 
Theoremlnopmuli 22544 Multiplicative property of a linear Hilbert space operator. (Contributed by NM, 11-May-2005.) (New usage is discouraged.)
 |-  T  e.  LinOp   =>    |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( T `  ( A  .h  B ) )  =  ( A  .h  ( T `  B ) ) )
 
Theoremlnopaddmuli 22545 Sum/product property of a linear Hilbert space operator. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.)
 |-  T  e.  LinOp   =>    |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( T `  ( B  +h  ( A  .h  C ) ) )  =  ( ( T `
  B )  +h  ( A  .h  ( T `  C ) ) ) )
 
Theoremlnopsubi 22546 Subtraction property for a linear Hilbert space operator. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.)
 |-  T  e.  LinOp   =>    |-  ( ( A  e.  ~H 
 /\  B  e.  ~H )  ->  ( T `  ( A  -h  B ) )  =  ( ( T `  A )  -h  ( T `  B ) ) )
 
Theoremlnopsubmuli 22547 Subtraction/product property of a linear Hilbert space operator. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)
 |-  T  e.  LinOp   =>    |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( T `  ( B  -h  ( A  .h  C ) ) )  =  ( ( T `
  B )  -h  ( A  .h  ( T `  C ) ) ) )
 
Theoremlnopmulsubi 22548 Product/subtraction property of a linear Hilbert space operator. (Contributed by NM, 2-Jul-2005.) (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 ) ) )
 
Theoremhomco2 22549 Move a scalar product out of a composition of operators. The operator  T must be linear, unlike homco1 22373 that works for any operators. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T  e.  LinOp  /\  U : ~H --> ~H )  ->  ( T  o.  ( A  .op  U ) )  =  ( A  .op  ( T  o.  U ) ) )
 
Theoremidunop 22550 The identity function (restricted to Hilbert space) is a unitary operator. (Contributed by NM, 21-Jan-2006.) (New usage is discouraged.)
 |-  (  _I  |`  ~H )  e. 
 UniOp
 
Theorem0cnop 22551 The identically zero function is a continuous Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
 |-  0hop  e. 
 ConOp
 
Theorem0cnfn 22552 The identically zero function is a continuous Hilbert space functional. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
 |-  ( ~H  X.  { 0 } )  e.  ConFn
 
Theoremidcnop 22553 The identity function (restricted to Hilbert space) is a continuous operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
 |-  (  _I  |`  ~H )  e. 
 ConOp
 
Theoremidhmop 22554 The Hilbert space identity operator is a Hermitian operator. (Contributed by NM, 22-Apr-2006.) (New usage is discouraged.)
 |-  Iop  e. 
 HrmOp
 
Theorem0hmop 22555 The identically zero function is a Hermitian operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)
 |-  0hop  e. 
 HrmOp
 
Theorem0lnop 22556 The identically zero function is a linear Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
 |-  0hop  e. 
 LinOp
 
Theorem0lnfn 22557 The identically zero function is a linear Hilbert space functional. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
 |-  ( ~H  X.  { 0 } )  e.  LinFn
 
Theoremnmop0 22558 The norm of the zero operator is zero. (Contributed by NM, 8-Feb-2006.) (New usage is discouraged.)
 |-  ( normop ` 
 0hop )  =  0
 
Theoremnmfn0 22559 The norm of the identically zero functional is zero. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
 |-  ( normfn `
  ( ~H  X.  { 0 } ) )  =  0
 
TheoremhmopbdoptHIL 22560 A Hermitian operator is a bounded linear operator (Hellinger-Toeplitz Theorem). (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.)
 |-  ( T  e.  HrmOp  ->  T  e. 
 BndLinOp )
 
Theoremhoddii 22561 Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 22352 does not require linearity.) (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
 |-  R  e.  LinOp   &    |-  S : ~H --> ~H   &    |-  T : ~H --> ~H   =>    |-  ( R  o.  ( S  -op  T ) )  =  ( ( R  o.  S )  -op  ( R  o.  T ) )
 
Theoremhoddi 22562 Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 22352 does not require linearity.) (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( R  e.  LinOp  /\  S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( R  o.  ( S  -op  T ) )  =  ( ( R  o.  S )  -op  ( R  o.  T ) ) )
 
Theoremnmop0h 22563 The norm of any operator on the trivial Hilbert space is zero. (This is the reason we need  ~H  =/=  0H in nmopun 22586.) (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( ~H  =  0H  /\  T : ~H --> ~H )  ->  ( normop `  T )  =  0 )
 
Theoremidlnop 22564 The identity function (restricted to Hilbert space) is a linear operator. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
 |-  (  _I  |`  ~H )  e. 
 LinOp
 
Theorem0bdop 22565 The identically zero operator is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
 |-  0hop  e.  BndLinOp
 
Theoremadj0 22566 Adjoint of the zero operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
 |-  ( adjh `  0hop )  =  0hop
 
Theoremnmlnop0iALT 22567 A linear operator with a zero norm is identically zero. (Contributed by NM, 8-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   =>    |-  ( ( normop `  T )  =  0  <->  T  =  0hop )
 
Theoremnmlnop0iHIL 22568 A linear operator with a zero norm is identically zero. (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.)
 |-  T  e.  LinOp   =>    |-  ( ( normop `  T )  =  0  <->  T  =  0hop )
 
Theoremnmlnopgt0i 22569 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  e.  LinOp   =>    |-  ( T  =/=  0hop  <->  0  <  ( normop `  T )
 )
 
Theoremnmlnop0 22570 A linear operator with a zero norm is identically zero. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
 |-  ( T  e.  LinOp  ->  (
 ( normop `  T )  =  0  <->  T  =  0hop ) )
 
Theoremnmlnopne0 22571 A linear operator with a nonzero norm is nonzero. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
 |-  ( T  e.  LinOp  ->  (
 ( normop `  T )  =/=  0  <->  T  =/=  0hop )
 )
 
Theoremlnopmi 22572 The scalar product of a linear operator is a linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   =>    |-  ( A  e.  CC  ->  ( A  .op  T )  e.  LinOp )
 
Theoremlnophsi 22573 The sum of two linear operators is linear. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  S  e.  LinOp   &    |-  T  e.  LinOp   =>    |-  ( S  +op  T )  e.  LinOp
 
Theoremlnophdi 22574 The difference of two linear operators is linear. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.)
 |-  S  e.  LinOp   &    |-  T  e.  LinOp   =>    |-  ( S  -op  T )  e.  LinOp
 
Theoremlnopcoi 22575 The composition of two linear operators is linear. (Contributed by NM, 8-Mar-2006.) (New usage is discouraged.)
 |-  S  e.  LinOp   &    |-  T  e.  LinOp   =>    |-  ( S  o.  T )  e.  LinOp
 
Theoremlnopco0i 22576 The composition of a linear operator with one whose norm is zero. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  S  e.  LinOp   &    |-  T  e.  LinOp   =>    |-  ( ( normop `  T )  =  0  ->  ( normop `
  ( S  o.  T ) )  =  0 )
 
Theoremlnopeq0lem1 22577 Lemma for lnopeq0i 22579. Apply the generalized polarization identity polid2i 21728 to the quadratic form  (
( T `  x
) ,  x ). (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( ( T `  A )  .ih  B )  =  ( ( ( ( ( T `  ( A  +h  B ) )  .ih  ( A  +h  B ) )  -  ( ( T `  ( A  -h  B ) )  .ih  ( A  -h  B ) ) )  +  ( _i  x.  ( ( ( T `
  ( A  +h  ( _i  .h  B ) ) )  .ih  ( A  +h  ( _i  .h  B ) ) )  -  ( ( T `  ( A  -h  ( _i  .h  B ) ) ) 
 .ih  ( A  -h  ( _i  .h  B ) ) ) ) ) )  /  4
 )
 
Theoremlnopeq0lem2 22578 Lemma for lnopeq0i 22579. (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   =>    |-  ( ( A  e.  ~H 
 /\  B  e.  ~H )  ->  ( ( T `
  A )  .ih  B )  =  ( ( ( ( ( T `
  ( A  +h  B ) )  .ih  ( A  +h  B ) )  -  ( ( T `  ( A  -h  B ) ) 
 .ih  ( A  -h  B ) ) )  +  ( _i  x.  ( ( ( T `
  ( A  +h  ( _i  .h  B ) ) )  .ih  ( A  +h  ( _i  .h  B ) ) )  -  ( ( T `  ( A  -h  ( _i  .h  B ) ) ) 
 .ih  ( A  -h  ( _i  .h  B ) ) ) ) ) )  /  4
 ) )
 
Theoremlnopeq0i 22579* A condition implying that a linear Hilbert space operator is identically zero. Unlike ho01i 22400 for arbitrary operators, when the operator is linear we need to consider only the values of the quadratic form  ( T `  x )  .ih  x
). (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   =>    |-  ( A. x  e. 
 ~H  ( ( T `
  x )  .ih  x )  =  0  <->  T  =  0hop )
 
Theoremlnopeqi 22580* Two linear Hilbert space operators are equal iff their quadratic forms are equal. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  U  e.  LinOp   =>    |-  ( A. x  e. 
 ~H  ( ( T `
  x )  .ih  x )  =  ( ( U `  x ) 
 .ih  x )  <->  T  =  U )
 
Theoremlnopeq 22581* Two linear Hilbert space operators are equal iff their quadratic forms are equal. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  LinOp  /\  U  e.  LinOp )  ->  ( A. x  e.  ~H  ( ( T `  x )  .ih  x )  =  ( ( U `
  x )  .ih  x )  <->  T  =  U ) )
 
Theoremlnopunilem1 22582* Lemma for lnopunii 22584. (Contributed by NM, 14-May-2005.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |- 
 A. x  e.  ~H  ( normh `  ( T `  x ) )  =  ( normh `  x )   &    |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  C  e.  CC   =>    |-  ( Re `  ( C  x.  ( ( T `
  A )  .ih  ( T `  B ) ) ) )  =  ( Re `  ( C  x.  ( A  .ih  B ) ) )
 
Theoremlnopunilem2 22583* Lemma for lnopunii 22584. (Contributed by NM, 12-May-2005.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |- 
 A. x  e.  ~H  ( normh `  ( T `  x ) )  =  ( normh `  x )   &    |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  (
 ( T `  A )  .ih  ( T `  B ) )  =  ( A  .ih  B )
 
Theoremlnopunii 22584* If a linear operator (whose range is  ~H) is idempotent in the norm, the operator is unitary. Similar to theorem in [AkhiezerGlazman] p. 73. (Contributed by NM, 23-Jan-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T : ~H -onto-> ~H   &    |-  A. x  e.  ~H  ( normh `  ( T `  x ) )  =  ( normh `  x )   =>    |-  T  e.  UniOp
 
Theoremelunop2 22585* An operator is unitary iff it is linear, onto, and idempotent in the norm. Similar to theorem in [AkhiezerGlazman] p. 73, and its converse. (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  UniOp  <->  ( T  e.  LinOp  /\  T : ~H -onto-> ~H  /\ 
 A. x  e.  ~H  ( normh `  ( T `  x ) )  =  ( normh `  x )
 ) )
 
Theoremnmopun 22586 Norm of a unitary Hilbert space operator. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( ~H  =/=  0H  /\  T  e.  UniOp )  ->  ( normop `  T )  =  1 )
 
Theoremunopbd 22587 A unitary operator is a bounded linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  ( T  e.  UniOp  ->  T  e. 
 BndLinOp )
 
Theoremlnophmlem1 22588* Lemma for lnophmi 22590. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  T  e.  LinOp   &    |- 
 A. x  e.  ~H  ( x  .ih  ( T `
  x ) )  e.  RR   =>    |-  ( A  .ih  ( T `  A ) )  e.  RR
 
Theoremlnophmlem2 22589* Lemma for lnophmi 22590. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  T  e.  LinOp   &    |- 
 A. x  e.  ~H  ( x  .ih  ( T `
  x ) )  e.  RR   =>    |-  ( A  .ih  ( T `  B ) )  =  ( ( T `
  A )  .ih  B )
 
Theoremlnophmi 22590* A linear operator is Hermitian if  x  .ih  ( T `
 x ) takes only real values. Remark in [ReedSimon] p. 195. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |- 
 A. x  e.  ~H  ( x  .ih  ( T `
  x ) )  e.  RR   =>    |-  T  e.  HrmOp
 
Theoremlnophm 22591* A linear operator is Hermitian if  x  .ih  ( T `
 x ) takes only real values. Remark in [ReedSimon] p. 195. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  LinOp  /\ 
 A. x  e.  ~H  ( x  .ih  ( T `
  x ) )  e.  RR )  ->  T  e.  HrmOp )
 
Theoremhmops 22592 The sum of two Hermitian operators is Hermitian. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  HrmOp  /\  U  e.  HrmOp )  ->  ( T  +op  U )  e.  HrmOp )
 
Theoremhmopm 22593 The scalar product of a Hermitian operator with a real is Hermitian. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  RR  /\  T  e.  HrmOp )  ->  ( A  .op  T )  e.  HrmOp )
 
Theoremhmopd 22594 The difference of two Hermitian operators is Hermitian. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  HrmOp  /\  U  e.  HrmOp )  ->  ( T  -op  U )  e.  HrmOp )
 
Theoremhmopco 22595 The composition of two commuting Hermitian operators is Hermitian. (Contributed by NM, 22-Aug-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  HrmOp  /\  U  e.  HrmOp  /\  ( T  o.  U )  =  ( U  o.  T ) )  ->  ( T  o.  U )  e. 
 HrmOp )
 
Theoremnmbdoplbi 22596 A lower bound for the norm of a bounded linear operator. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
 |-  T  e. 
 BndLinOp   =>    |-  ( A  e.  ~H  ->  ( normh `  ( T `  A ) )  <_  ( ( normop `  T )  x.  ( normh `  A ) ) )
 
Theoremnmbdoplb 22597 A lower bound for the norm of a bounded linear Hilbert space operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  BndLinOp  /\  A  e.  ~H )  ->  ( normh `  ( T `  A ) )  <_  ( ( normop `  T )  x.  ( normh `  A ) ) )
 
Theoremnmcexi 22598* Lemma for nmcopexi 22599 and nmcfnexi 22623. The norm of a continuous linear Hilbert space operator or functional exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by Mario Carneiro, 17-Nov-2013.) (Proof shortened by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  E. y  e.  RR+  A. z  e.  ~H  ( ( normh `  z
 )  <  y  ->  ( N `  ( T `
  z ) )  <  1 )   &    |-  ( S `  T )  = 
 sup ( { m  |  E. x  e.  ~H  ( ( normh `  x )  <_  1  /\  m  =  ( N `  ( T `  x ) ) ) } ,  RR* ,  <  )   &    |-  ( x  e. 
 ~H  ->  ( N `  ( T `  x ) )  e.  RR )   &    |-  ( N `  ( T `  0h ) )  =  0   &    |-  ( ( ( y 
 /  2 )  e.  RR+  /\  x  e.  ~H )  ->  ( ( y 
 /  2 )  x.  ( N `  ( T `  x ) ) )  =  ( N `
  ( T `  ( ( y  / 
 2 )  .h  x ) ) ) )   =>    |-  ( S `  T )  e.  RR
 
Theoremnmcopexi 22599 The norm of a continuous linear Hilbert space operator exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by NM, 5-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T  e.  ConOp   =>    |-  ( normop `  T )  e.  RR
 
Theoremnmcoplbi 22600 A lower bound for the norm of a continuous linear operator. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 7-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T  e.  ConOp   =>    |-  ( A  e.  ~H  ->  ( normh `  ( T `  A ) )  <_  ( ( normop `  T )  x.  ( normh `  A ) ) )
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