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Theorem List for Metamath Proof Explorer - 22401-22500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnmop0h 22401 The norm of any operator on the trivial Hilbert space is zero. (This is the reason we need  ~H  =/=  0H in nmopun 22424.) (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( ~H  =  0H  /\  T : ~H --> ~H )  ->  ( normop `  T )  =  0 )
 
Theoremidlnop 22402 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 22403 The identically zero operator is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
 |-  0hop  e.  BndLinOp
 
Theoremadj0 22404 Adjoint of the zero operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
 |-  ( adjh `  0hop )  =  0hop
 
Theoremnmlnop0iALT 22405 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 22406 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 22407 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 22408 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 22409 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 22410 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 22411 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 22412 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 22413 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 22414 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 22415 Lemma for lnopeq0i 22417. Apply the generalized polarization identity polid2i 21566 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 22416 Lemma for lnopeq0i 22417. (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 22417* A condition implying that a linear Hilbert space operator is identically zero. Unlike ho01i 22238 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 22418* 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 22419* 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 22420* Lemma for lnopunii 22422. (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 22421* Lemma for lnopunii 22422. (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 22422* 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 22423* 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 22424 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 22425 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 22426* Lemma for lnophmi 22428. (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 22427* Lemma for lnophmi 22428. (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 22428* 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 22429* 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 22430 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 22431 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 22432 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 22433 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 22434 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 22435 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 22436* Lemma for nmcopexi 22437 and nmcfnexi 22461. 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 22437 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 22438 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 22439 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 22440 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 22441 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 22442 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 22443* Lemma for lnopconi 22444 and lnfnconi 22465. (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 22444* 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 22445* 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 22446 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 22447 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 22448 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 22449 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 22450 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 22451 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 22452 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 22453 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 22454 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 22455 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 22456 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 22457 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 22458 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 22459 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 22460 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 22461 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 22462 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 22463 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 22464 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 22465* 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 22466* 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 22467 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 22468 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 22469 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 22470 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 22471 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
 
15.9.42  Riesz lemma
 
Theoremriesz3i 22472* 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 22473* 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 22474* A continuous linear functional can be expressed as an inner product. Uniqueness part of Theorem 3.9 of [Beran] p. 104. See riesz2 22476 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 22475* 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 22476. For the continuous linear functional version, see riesz3i 22472 and riesz4 22474. (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 22476* 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 22475. (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 ) )
 
15.9.43  Adjoints (cont.)
 
Theoremcnlnadjlem1 22477* Lemma for cnlnadji 22486 (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 22478* Lemma for cnlnadji 22486. 
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 22479* Lemma for cnlnadji 22486. By riesz4 22474, 
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 22480* Lemma for cnlnadji 22486. 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 22481* Lemma for cnlnadji 22486. 
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 22482* Lemma for cnlnadji 22486. 
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 22483* Lemma for cnlnadji 22486. 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 22484* Lemma for cnlnadji 22486. 
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 22485* Lemma for cnlnadji 22486. 
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 22486* 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 22487* 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 22488* 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 22489* 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 22490 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 22491 Every bounded linear Hilbert space operator has an adjoint. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
 |-  BndLinOp  C_  dom  adjh
 
Theorembdopadj 22492 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 22493 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 22494 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 22495 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 22496 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 22497 Every operator with an adjoint is linear. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.)
 |-  dom  adjh  C_  LinOp
 
Theoremnmopadjlei 22498 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 22499 Lemma for nmopadji 22500. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)
 |-  T  e. 
 BndLinOp   =>    |-  ( normop `  ( adjh `  T ) )  <_  ( normop `  T )
 
Theoremnmopadji 22500 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 )
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