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Theorem List for Metamath Proof Explorer - 22601-22700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlnopeq0lem1 22601 Lemma for lnopeq0i 22603. Apply the generalized polarization identity polid2i 21752 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 22602 Lemma for lnopeq0i 22603. (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 22603* A condition implying that a linear Hilbert space operator is identically zero. Unlike ho01i 22424 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 22604* 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 22605* 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 22606* Lemma for lnopunii 22608. (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 22607* Lemma for lnopunii 22608. (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 22608* 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 22609* 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 22610 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 22611 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 22612* Lemma for lnophmi 22614. (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 22613* Lemma for lnophmi 22614. (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 22614* 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 22615* 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 22616 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 22617 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 22618 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 22619 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 22620 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 22621 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 22622* Lemma for nmcopexi 22623 and nmcfnexi 22647. 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 22623 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 22624 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 ) ) )
 
Theoremnmcopex 22625 The norm of a continuous linear Hilbert space operator exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  LinOp  /\  T  e.  ConOp )  ->  ( normop `  T )  e.  RR )
 
Theoremnmcoplb 22626 A lower bound for the norm of a continuous linear Hilbert space operator. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  LinOp  /\  T  e.  ConOp  /\  A  e.  ~H )  ->  ( normh `  ( T `  A ) )  <_  ( ( normop `  T )  x.  ( normh `  A ) ) )
 
Theoremnmophmi 22627 The norm of the scalar product of a bounded linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  T  e. 
 BndLinOp   =>    |-  ( A  e.  CC  ->  ( normop `  ( A  .op  T ) )  =  ( ( abs `  A )  x.  ( normop `  T ) ) )
 
Theorembdophmi 22628 The scalar product of a bounded linear operator is a bounded linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  T  e. 
 BndLinOp   =>    |-  ( A  e.  CC  ->  ( A  .op  T )  e.  BndLinOp )
 
Theoremlnconi 22629* Lemma for lnopconi 22630 and lnfnconi 22651. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  C  ->  S  e.  RR )   &    |-  (
 ( T  e.  C  /\  y  e.  ~H )  ->  ( N `  ( T `  y ) )  <_  ( S  x.  ( normh `  y )
 ) )   &    |-  ( T  e.  C 
 <-> 
 A. x  e.  ~H  A. z  e.  RR+  E. y  e.  RR+  A. w  e.  ~H  ( ( normh `  ( w  -h  x ) )  <  y  ->  ( N `  ( ( T `
  w ) M ( T `  x ) ) )  < 
 z ) )   &    |-  (
 y  e.  ~H  ->  ( N `  ( T `
  y ) )  e.  RR )   &    |-  (
 ( w  e.  ~H  /\  x  e.  ~H )  ->  ( T `  ( w  -h  x ) )  =  ( ( T `
  w ) M ( T `  x ) ) )   =>    |-  ( T  e.  C 
 <-> 
 E. x  e.  RR  A. y  e.  ~H  ( N `  ( T `  y ) )  <_  ( x  x.  ( normh `  y ) ) )
 
Theoremlnopconi 22630* A condition equivalent to " T is continuous" when 
T is linear. Theorem 3.5(iii) of [Beran] p. 99. (Contributed by NM, 7-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
 |-  T  e.  LinOp   =>    |-  ( T  e.  ConOp  <->  E. x  e.  RR  A. y  e.  ~H  ( normh `  ( T `  y ) ) 
 <_  ( x  x.  ( normh `  y ) ) )
 
Theoremlnopcon 22631* A condition equivalent to " T is continuous" when 
T is linear. Theorem 3.5(iii) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  LinOp  ->  ( T  e.  ConOp  <->  E. x  e.  RR  A. y  e.  ~H  ( normh `  ( T `  y ) )  <_  ( x  x.  ( normh `  y ) ) ) )
 
Theoremlnopcnbd 22632 A linear operator is continuous iff it is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  LinOp  ->  ( T  e.  ConOp  <->  T  e.  BndLinOp ) )
 
Theoremlncnopbd 22633 A continuous linear operator is a bounded linear operator. This theorem justifies our use of "bounded linear" as an interchangeable condition for "continuous linear" used in some textbook proofs. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  ( LinOp  i^i  ConOp )  <->  T  e.  BndLinOp )
 
Theoremlncnbd 22634 A continuous linear operator is a bounded linear operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
 |-  ( LinOp  i^i  ConOp )  =  BndLinOp
 
Theoremlnopcnre 22635 A linear operator is continuous iff it is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  LinOp  ->  ( T  e.  ConOp  <->  ( normop `  T )  e.  RR )
 )
 
Theoremlnfnli 22636 Basic property of a linear Hilbert space 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 ) ) )
 
Theoremlnfnfi 22637 A linear Hilbert space functional is a functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinFn   =>    |-  T : ~H --> CC
 
Theoremlnfn0i 22638 The value of a linear Hilbert space functional at zero is zero. Remark in [Beran] p. 99. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinFn   =>    |-  ( T `  0h )  =  0
 
Theoremlnfnaddi 22639 Additive property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinFn   =>    |-  ( ( A  e.  ~H 
 /\  B  e.  ~H )  ->  ( T `  ( A  +h  B ) )  =  ( ( T `  A )  +  ( T `  B ) ) )
 
Theoremlnfnmuli 22640 Multiplicative property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinFn   =>    |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( T `  ( A  .h  B ) )  =  ( A  x.  ( T `  B ) ) )
 
Theoremlnfnaddmuli 22641 Sum/product property of a linear Hilbert space functional. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinFn   =>    |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( T `  ( B  +h  ( A  .h  C ) ) )  =  ( ( T `
  B )  +  ( A  x.  ( T `  C ) ) ) )
 
Theoremlnfnsubi 22642 Subtraction property for a linear Hilbert space functional. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinFn   =>    |-  ( ( A  e.  ~H 
 /\  B  e.  ~H )  ->  ( T `  ( A  -h  B ) )  =  ( ( T `  A )  -  ( T `  B ) ) )
 
Theoremlnfn0 22643 The value of a linear Hilbert space functional at zero is zero. Remark in [Beran] p. 99. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
 |-  ( T  e.  LinFn  ->  ( T `  0h )  =  0 )
 
Theoremlnfnmul 22644 Multiplicative property of a linear Hilbert space functional. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  LinFn  /\  A  e.  CC  /\  B  e.  ~H )  ->  ( T `  ( A  .h  B ) )  =  ( A  x.  ( T `  B ) ) )
 
Theoremnmbdfnlbi 22645 A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
 |-  ( T  e.  LinFn  /\  ( normfn `
  T )  e. 
 RR )   =>    |-  ( A  e.  ~H  ->  ( abs `  ( T `  A ) ) 
 <_  ( ( normfn `  T )  x.  ( normh `  A ) ) )
 
Theoremnmbdfnlb 22646 A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  LinFn  /\  ( normfn `  T )  e.  RR  /\  A  e.  ~H )  ->  ( abs `  ( T `  A ) )  <_  ( (
 normfn `  T )  x.  ( normh `  A )
 ) )
 
Theoremnmcfnexi 22647 The norm of a continuous linear Hilbert space functional exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
 |-  T  e.  LinFn   &    |-  T  e.  ConFn   =>    |-  ( normfn `  T )  e.  RR
 
Theoremnmcfnlbi 22648 A lower bound for the norm of a continuous linear functional. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinFn   &    |-  T  e.  ConFn   =>    |-  ( A  e.  ~H  ->  ( abs `  ( T `  A ) ) 
 <_  ( ( normfn `  T )  x.  ( normh `  A ) ) )
 
Theoremnmcfnex 22649 The norm of a continuous linear Hilbert space functional exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  LinFn  /\  T  e.  ConFn )  ->  ( normfn `  T )  e.  RR )
 
Theoremnmcfnlb 22650 A lower bound of the norm of a continuous linear Hilbert space functional. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  LinFn  /\  T  e.  ConFn  /\  A  e.  ~H )  ->  ( abs `  ( T `  A ) )  <_  ( ( normfn `  T )  x.  ( normh `  A ) ) )
 
Theoremlnfnconi 22651* A condition equivalent to " T is continuous" when 
T is linear. Theorem 3.5(iii) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
 |-  T  e.  LinFn   =>    |-  ( T  e.  ConFn  <->  E. x  e.  RR  A. y  e.  ~H  ( abs `  ( T `  y ) ) 
 <_  ( x  x.  ( normh `  y ) ) )
 
Theoremlnfncon 22652* A condition equivalent to " T is continuous" when 
T is linear. Theorem 3.5(iii) of [Beran] p. 99. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  LinFn  ->  ( T  e.  ConFn  <->  E. x  e.  RR  A. y  e.  ~H  ( abs `  ( T `  y ) )  <_  ( x  x.  ( normh `  y ) ) ) )
 
Theoremlnfncnbd 22653 A linear functional is continuous iff it is bounded. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
 |-  ( T  e.  LinFn  ->  ( T  e.  ConFn  <->  ( normfn `  T )  e.  RR )
 )
 
Theoremimaelshi 22654 The image of a subspace under a linear operator is a subspace. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  A  e.  SH   =>    |-  ( T " A )  e.  SH
 
Theoremrnelshi 22655 The range of a linear operator is a subspace. (Contributed by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
 |-  T  e.  LinOp   =>    |- 
 ran  T  e.  SH
 
Theoremnlelshi 22656 The null space of a linear functional is a subspace. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
 |-  T  e.  LinFn   =>    |-  ( null `  T )  e.  SH
 
Theoremnlelchi 22657 The null space of a continuous linear functional is a closed subspace. Remark 3.8 of [Beran] p. 103. (Contributed by NM, 11-Feb-2006.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  T  e.  LinFn   &    |-  T  e.  ConFn   =>    |-  ( null `  T )  e.  CH
 
17.6.11  Riesz lemma
 
Theoremriesz3i 22658* A continuous linear functional can be expressed as an inner product. Existence part of Theorem 3.9 of [Beran] p. 104. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinFn   &    |-  T  e.  ConFn   =>    |- 
 E. w  e.  ~H  A. v  e.  ~H  ( T `  v )  =  ( v  .ih  w )
 
Theoremriesz4i 22659* A continuous linear functional can be expressed as an inner product. Uniqueness part of Theorem 3.9 of [Beran] p. 104. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinFn   &    |-  T  e.  ConFn   =>    |- 
 E! w  e.  ~H  A. v  e.  ~H  ( T `  v )  =  ( v  .ih  w )
 
Theoremriesz4 22660* A continuous linear functional can be expressed as an inner product. Uniqueness part of Theorem 3.9 of [Beran] p. 104. See riesz2 22662 for the bounded linear functional version. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  ( LinFn  i^i  ConFn )  ->  E! w  e.  ~H  A. v  e. 
 ~H  ( T `  v )  =  (
 v  .ih  w )
 )
 
Theoremriesz1 22661* Part 1 of the Riesz representation theorem for bounded linear functionals. A linear functional is bounded iff its value can be expressed as an inner product. Part of Theorem 17.3 of [Halmos] p. 31. For part 2, see riesz2 22662. For the continuous linear functional version, see riesz3i 22658 and riesz4 22660. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
 |-  ( T  e.  LinFn  ->  (
 ( normfn `  T )  e.  RR  <->  E. y  e.  ~H  A. x  e.  ~H  ( T `  x )  =  ( x  .ih  y
 ) ) )
 
Theoremriesz2 22662* Part 2 of the Riesz representation theorem for bounded linear functionals. The value of a bounded linear functional corresponds to a unique inner product. Part of Theorem 17.3 of [Halmos] p. 31. For part 1, see riesz1 22661. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( T  e.  LinFn  /\  ( normfn `  T )  e.  RR )  ->  E! y  e.  ~H  A. x  e.  ~H  ( T `  x )  =  ( x  .ih  y ) )
 
17.6.12  Adjoints (cont.)
 
Theoremcnlnadjlem1 22663* Lemma for cnlnadji 22672 (Theorem 3.10 of [Beran] p. 104: every continuous linear operator has an adjoint). The value of the auxiliary functional  G. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T  e.  ConOp   &    |-  G  =  ( g  e.  ~H  |->  ( ( T `  g ) 
 .ih  y ) )   =>    |-  ( A  e.  ~H  ->  ( G `  A )  =  ( ( T `  A )  .ih  y ) )
 
Theoremcnlnadjlem2 22664* Lemma for cnlnadji 22672. 
G is a continuous linear functional. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T  e.  ConOp   &    |-  G  =  ( g  e.  ~H  |->  ( ( T `  g ) 
 .ih  y ) )   =>    |-  ( y  e.  ~H  ->  ( G  e.  LinFn  /\  G  e.  ConFn ) )
 
Theoremcnlnadjlem3 22665* Lemma for cnlnadji 22672. By riesz4 22660, 
B is the unique vector such that  ( T `  v )  .ih  y
)  =  ( v 
.ih  w ) for all  v. (Contributed by NM, 17-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T  e.  ConOp   &    |-  G  =  ( g  e.  ~H  |->  ( ( T `  g ) 
 .ih  y ) )   &    |-  B  =  ( iota_ w  e. 
 ~H A. v  e.  ~H  ( ( T `  v )  .ih  y )  =  ( v  .ih  w ) )   =>    |-  ( y  e.  ~H  ->  B  e.  ~H )
 
Theoremcnlnadjlem4 22666* Lemma for cnlnadji 22672. The values of auxiliary function  F are vectors. (Contributed by NM, 17-Feb-2006.) (Proof shortened by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T  e.  ConOp   &    |-  G  =  ( g  e.  ~H  |->  ( ( T `  g ) 
 .ih  y ) )   &    |-  B  =  ( iota_ w  e. 
 ~H A. v  e.  ~H  ( ( T `  v )  .ih  y )  =  ( v  .ih  w ) )   &    |-  F  =  ( y  e.  ~H  |->  B )   =>    |-  ( A  e.  ~H  ->  ( F `  A )  e.  ~H )
 
Theoremcnlnadjlem5 22667* Lemma for cnlnadji 22672. 
F is an adjoint of  T (later, we will show it is unique). (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T  e.  ConOp   &    |-  G  =  ( g  e.  ~H  |->  ( ( T `  g ) 
 .ih  y ) )   &    |-  B  =  ( iota_ w  e. 
 ~H A. v  e.  ~H  ( ( T `  v )  .ih  y )  =  ( v  .ih  w ) )   &    |-  F  =  ( y  e.  ~H  |->  B )   =>    |-  ( ( A  e.  ~H 
 /\  C  e.  ~H )  ->  ( ( T `
  C )  .ih  A )  =  ( C 
 .ih  ( F `  A ) ) )
 
Theoremcnlnadjlem6 22668* Lemma for cnlnadji 22672. 
F is linear. (Contributed by NM, 17-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T  e.  ConOp   &    |-  G  =  ( g  e.  ~H  |->  ( ( T `  g ) 
 .ih  y ) )   &    |-  B  =  ( iota_ w  e. 
 ~H A. v  e.  ~H  ( ( T `  v )  .ih  y )  =  ( v  .ih  w ) )   &    |-  F  =  ( y  e.  ~H  |->  B )   =>    |-  F  e.  LinOp
 
Theoremcnlnadjlem7 22669* Lemma for cnlnadji 22672. Helper lemma to show that  F is continuous. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T  e.  ConOp   &    |-  G  =  ( g  e.  ~H  |->  ( ( T `  g ) 
 .ih  y ) )   &    |-  B  =  ( iota_ w  e. 
 ~H A. v  e.  ~H  ( ( T `  v )  .ih  y )  =  ( v  .ih  w ) )   &    |-  F  =  ( y  e.  ~H  |->  B )   =>    |-  ( A  e.  ~H  ->  ( normh `  ( F `  A ) )  <_  ( ( normop `  T )  x.  ( normh `  A ) ) )
 
Theoremcnlnadjlem8 22670* Lemma for cnlnadji 22672. 
F is continuous. (Contributed by NM, 17-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T  e.  ConOp   &    |-  G  =  ( g  e.  ~H  |->  ( ( T `  g ) 
 .ih  y ) )   &    |-  B  =  ( iota_ w  e. 
 ~H A. v  e.  ~H  ( ( T `  v )  .ih  y )  =  ( v  .ih  w ) )   &    |-  F  =  ( y  e.  ~H  |->  B )   =>    |-  F  e.  ConOp
 
Theoremcnlnadjlem9 22671* Lemma for cnlnadji 22672. 
F provides an example showing the existence of a continuous linear adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T  e.  ConOp   &    |-  G  =  ( g  e.  ~H  |->  ( ( T `  g ) 
 .ih  y ) )   &    |-  B  =  ( iota_ w  e. 
 ~H A. v  e.  ~H  ( ( T `  v )  .ih  y )  =  ( v  .ih  w ) )   &    |-  F  =  ( y  e.  ~H  |->  B )   =>    |- 
 E. t  e.  ( LinOp  i^i  ConOp ) A. x  e.  ~H  A. z  e. 
 ~H  ( ( T `
  x )  .ih  z )  =  ( x  .ih  ( t `  z ) )
 
Theoremcnlnadji 22672* Every continuous linear operator has an adjoint. Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T  e.  ConOp   =>    |- 
 E. t  e.  ( LinOp  i^i  ConOp ) A. x  e.  ~H  A. y  e. 
 ~H  ( ( T `
  x )  .ih  y )  =  ( x  .ih  ( t `  y ) )
 
Theoremcnlnadjeui 22673* Every continuous linear operator has a unique adjoint. Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
 |-  T  e.  LinOp   &    |-  T  e.  ConOp   =>    |- 
 E! t  e.  ( LinOp  i^i  ConOp ) A. x  e.  ~H  A. y  e. 
 ~H  ( ( T `
  x )  .ih  y )  =  ( x  .ih  ( t `  y ) )
 
Theoremcnlnadjeu 22674* Every continuous linear operator has a unique adjoint. Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  ( LinOp  i^i  ConOp )  ->  E! t  e.  ( LinOp  i^i  ConOp ) A. x  e.  ~H  A. y  e.  ~H  ( ( T `
  x )  .ih  y )  =  ( x  .ih  ( t `  y ) ) )
 
Theoremcnlnadj 22675* Every continuous linear operator has an adjoint. Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  ( LinOp  i^i  ConOp )  ->  E. t  e.  ( LinOp  i^i  ConOp ) A. x  e.  ~H  A. y  e.  ~H  ( ( T `
  x )  .ih  y )  =  ( x  .ih  ( t `  y ) ) )
 
Theoremcnlnssadj 22676 Every continuous linear Hilbert space operator has an adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
 |-  ( LinOp  i^i  ConOp )  C_  dom  adjh
 
Theorembdopssadj 22677 Every bounded linear Hilbert space operator has an adjoint. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
 |-  BndLinOp  C_  dom  adjh
 
Theorembdopadj 22678 Every bounded linear Hilbert space operator has an adjoint. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  BndLinOp  ->  T  e.  dom  adjh )
 
Theoremadjbdln 22679 The adjoint of a bounded linear operator is a bounded linear operator. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  BndLinOp  ->  ( adjh `  T )  e.  BndLinOp )
 
Theoremadjbdlnb 22680 An operator is bounded and linear iff its adjoint is. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
 |-  ( T  e.  BndLinOp  <->  ( adjh `  T )  e.  BndLinOp )
 
Theoremadjbd1o 22681 The mapping of adjoints of bounded linear operators is one-to-one onto. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
 |-  ( adjh 
 |`  BndLinOp ) : BndLinOp -1-1-onto->
 BndLinOp
 
Theoremadjlnop 22682 The adjoint of an operator is linear. Proposition 1 of [AkhiezerGlazman] p. 80. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.)
 |-  ( T  e.  dom  adjh  ->  (
 adjh `  T )  e. 
 LinOp )
 
Theoremadjsslnop 22683 Every operator with an adjoint is linear. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.)
 |-  dom  adjh  C_  LinOp
 
Theoremnmopadjlei 22684 Property of the norm of an adjoint. Part of proof of Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)
 |-  T  e. 
 BndLinOp   =>    |-  ( A  e.  ~H  ->  ( normh `  ( ( adjh `  T ) `  A ) )  <_  ( ( normop `  T )  x.  ( normh `  A ) ) )
 
Theoremnmopadjlem 22685 Lemma for nmopadji 22686. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)
 |-  T  e. 
 BndLinOp   =>    |-  ( normop `  ( adjh `  T ) )  <_  ( normop `  T )
 
Theoremnmopadji 22686 Property of the norm of an adjoint. Theorem 3.11(v) of [Beran] p. 106. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)
 |-  T  e. 
 BndLinOp   =>    |-  ( normop `  ( adjh `  T ) )  =  ( normop `  T )
 
Theoremadjeq0 22687 An operator is zero iff its adjoint is zero. Theorem 3.11(i) of [Beran] p. 106. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
 |-  ( T  =  0hop  <->  ( adjh `  T )  =  0hop )
 
Theoremadjmul 22688 The adjoint of the scalar product of an operator. Theorem 3.11(ii) of [Beran] p. 106. (Contributed by NM, 21-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T  e.  dom  adjh ) 
 ->  ( adjh `  ( A  .op  T ) )  =  ( ( * `  A )  .op  ( adjh `  T ) ) )
 
Theoremadjadd 22689 The adjoint of the sum of two operators. Theorem 3.11(iii) of [Beran] p. 106. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( S  e.  dom  adjh  /\  T  e.  dom  adjh ) 
 ->  ( adjh `  ( S  +op  T ) )  =  ( ( adjh `  S )  +op  ( adjh `  T ) ) )
 
Theoremnmoptrii 22690 Triangle inequality for the norms of bounded linear operators. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  S  e. 
 BndLinOp   &    |-  T  e.  BndLinOp   =>    |-  ( normop `  ( S  +op  T ) )  <_  ( ( normop `  S )  +  ( normop `  T ) )
 
Theoremnmopcoi 22691 Upper bound for the norm of the composition of two bounded linear operators. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  S  e. 
 BndLinOp   &    |-  T  e.  BndLinOp   =>    |-  ( normop `  ( S  o.  T ) )  <_  ( ( normop `  S )  x.  ( normop `  T ) )
 
Theorembdophsi 22692 The sum of two bounded linear operators is a bounded linear operator. (Contributed by NM, 9-Mar-2006.) (New usage is discouraged.)
 |-  S  e. 
 BndLinOp   &    |-  T  e.  BndLinOp   =>    |-  ( S  +op  T )  e.  BndLinOp
 
Theorembdophdi 22693 The difference between two bounded linear operators is bounded. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  S  e. 
 BndLinOp   &    |-  T  e.  BndLinOp   =>    |-  ( S  -op  T )  e.  BndLinOp
 
Theorembdopcoi 22694 The composition of two bounded linear operators is bounded. (Contributed by NM, 9-Mar-2006.) (New usage is discouraged.)
 |-  S  e. 
 BndLinOp   &    |-  T  e.  BndLinOp   =>    |-  ( S  o.  T )  e.  BndLinOp
 
Theoremnmoptri2i 22695 Triangle-type inequality for the norms of bounded linear operators. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  S  e. 
 BndLinOp   &    |-  T  e.  BndLinOp   =>    |-  ( ( normop `  S )  -  ( normop `  T ) )  <_  ( normop `  ( S  +op  T ) )
 
Theoremadjcoi 22696 The adjoint of a composition of bounded linear operators. Theorem 3.11(viii) of [Beran] p. 106. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  S  e. 
 BndLinOp   &    |-  T  e.  BndLinOp   =>    |-  ( adjh `  ( S  o.  T ) )  =  ( ( adjh `  T )  o.  ( adjh `  S ) )
 
Theoremnmopcoadji 22697 The norm of an operator composed with its adjoint. Part of Theorem 3.11(vi) of [Beran] p. 106. (Contributed by NM, 8-Mar-2006.) (New usage is discouraged.)
 |-  T  e. 
 BndLinOp   =>    |-  ( normop `  ( ( adjh `  T )  o.  T ) )  =  ( ( normop `  T ) ^ 2 )
 
Theoremnmopcoadj2i 22698 The norm of an operator composed with its adjoint. Part of Theorem 3.11(vi) of [Beran] p. 106. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  T  e. 
 BndLinOp   =>    |-  ( normop `  ( T  o.  ( adjh `  T )
 ) )  =  ( ( normop `  T ) ^ 2 )
 
Theoremnmopcoadj0i 22699 An operator composed with its adjoint is zero iff the operator is zero. Theorem 3.11(vii) of [Beran] p. 106. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  T  e. 
 BndLinOp   =>    |-  ( ( T  o.  ( adjh `  T )
 )  =  0hop  <->  T  =  0hop )
 
17.6.13  Quantum computation error bound theorem
 
Theoremunierri 22700 If we approximate a chain of unitary transformations (quantum computer gates)  F,  G by other unitary transformations  S,  T, the error increases at most additively. Equation 4.73 of [NielsenChuang] p. 195. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  F  e.  UniOp   &    |-  G  e.  UniOp   &    |-  S  e.  UniOp   &    |-  T  e.  UniOp   =>    |-  ( normop `  ( ( F  o.  G )  -op  ( S  o.  T ) ) )  <_  ( ( normop `  ( F  -op  S ) )  +  ( normop `  ( G  -op  T ) ) )
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