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Theorem List for Metamath Proof Explorer - 22201-22300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-0o 22201* Define the zero operator between two normed complex vector spaces. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
 |- 
 0op  =  ( u  e. 
 NrmCVec ,  w  e.  NrmCVec  |->  ( ( BaseSet `  u )  X.  { ( 0vec `  w ) } ) )
 
Syntaxcaj 22202 Adjoint of an operator.
 class  adj
 
Syntaxchmo 22203 Set of Hermitional (self-adjoint) operators.
 class  HmOp
 
Definitiondf-aj 22204* Define the adjoint of an operator (if it exists). The domain of  U adj W is the set of all operators from  U to  W that have an adjoint. Definition 3.9-1 of [Kreyszig] p. 196, although we don't require that  U and  W be Hilbert spaces nor that the operators be linear. Although we define it for any normed vector space for convenience, the definition is meaningful only for inner product spaces. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
 |- 
 adj  =  ( u  e. 
 NrmCVec ,  w  e.  NrmCVec  |->  {
 <. t ,  s >.  |  ( t : (
 BaseSet `  u ) --> ( BaseSet `  w )  /\  s : ( BaseSet `  w ) --> ( BaseSet `  u )  /\  A. x  e.  ( BaseSet `  u ) A. y  e.  ( BaseSet `  w )
 ( ( t `  x ) ( .i
 OLD `  w )
 y )  =  ( x ( .i OLD `  u ) ( s `
  y ) ) ) } )
 
Definitiondf-hmo 22205* Define the set of Hermitian (self-adjoint) operators on a normed complex vector space (normally a Hilbert space). Although we define it for any normed vector space for convenience, the definition is meaningful only for inner product spaces. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
 |- 
 HmOp  =  ( u  e. 
 NrmCVec 
 |->  { t  e.  dom  ( u adj u )  |  ( ( u adj u ) `  t )  =  t } )
 
Theoremlnoval 22206* The set of linear operators between two normed complex vector spaces. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  G  =  ( +v `  U )   &    |-  H  =  ( +v
 `  W )   &    |-  R  =  ( .s OLD `  U )   &    |-  S  =  ( .s
 OLD `  W )   &    |-  L  =  ( U  LnOp  W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  L  =  { t  e.  ( Y  ^m  X )  |  A. x  e. 
 CC  A. y  e.  X  A. z  e.  X  ( t `  ( ( x R y ) G z ) )  =  ( ( x S ( t `  y ) ) H ( t `  z
 ) ) } )
 
Theoremislno 22207* The predicate "is a linear operator." (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  G  =  ( +v `  U )   &    |-  H  =  ( +v
 `  W )   &    |-  R  =  ( .s OLD `  U )   &    |-  S  =  ( .s
 OLD `  W )   &    |-  L  =  ( U  LnOp  W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( T  e.  L  <->  ( T : X --> Y  /\  A. x  e.  CC  A. y  e.  X  A. z  e.  X  ( T `  ( ( x R y ) G z ) )  =  ( ( x S ( T `  y ) ) H ( T `
  z ) ) ) ) )
 
Theoremlnolin 22208 Basic linearity property of a linear operator. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  G  =  ( +v `  U )   &    |-  H  =  ( +v
 `  W )   &    |-  R  =  ( .s OLD `  U )   &    |-  S  =  ( .s
 OLD `  W )   &    |-  L  =  ( U  LnOp  W )   =>    |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L ) 
 /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  ->  ( T `  ( ( A R B ) G C ) )  =  (
 ( A S ( T `  B ) ) H ( T `
  C ) ) )
 
Theoremlnof 22209 A linear operator is a mapping. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 18-Nov-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  L  =  ( U  LnOp  W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  T : X --> Y )
 
Theoremlno0 22210 The value of a linear operator at zero is zero. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 18-Nov-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  Q  =  ( 0vec `  U )   &    |-  Z  =  ( 0vec `  W )   &    |-  L  =  ( U  LnOp  W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T `  Q )  =  Z )
 
Theoremlnocoi 22211 The composition of two linear operators is linear. (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
 |-  L  =  ( U 
 LnOp  W )   &    |-  M  =  ( W  LnOp  X )   &    |-  N  =  ( U  LnOp  X )   &    |-  U  e.  NrmCVec   &    |-  W  e.  NrmCVec   &    |-  X  e.  NrmCVec   &    |-  S  e.  L   &    |-  T  e.  M   =>    |-  ( T  o.  S )  e.  N
 
Theoremlnoadd 22212 Addition property of a linear operator. (Contributed by NM, 7-Dec-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  H  =  ( +v `  W )   &    |-  L  =  ( U  LnOp  W )   =>    |-  (
 ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
 )  ->  ( T `  ( A G B ) )  =  (
 ( T `  A ) H ( T `  B ) ) )
 
Theoremlnosub 22213 Subtraction property of a linear operator. (Contributed by NM, 7-Dec-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  ( -v `  W )   &    |-  L  =  ( U  LnOp  W )   =>    |-  (
 ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
 )  ->  ( T `  ( A M B ) )  =  (
 ( T `  A ) N ( T `  B ) ) )
 
Theoremlnomul 22214 Scalar multiplication property of a linear operator. (Contributed by NM, 5-Dec-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  R  =  ( .s OLD `  U )   &    |-  S  =  ( .s
 OLD `  W )   &    |-  L  =  ( U  LnOp  W )   =>    |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L ) 
 /\  ( A  e.  CC  /\  B  e.  X ) )  ->  ( T `
  ( A R B ) )  =  ( A S ( T `  B ) ) )
 
Theoremnvo00 22215 Two ways to express a zero operator. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  T : X --> Y ) 
 ->  ( T  =  ( X  X.  { Z } )  <->  ran  T  =  { Z } ) )
 
Theoremnmoofval 22216* The operator norm function. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  L  =  ( normCV `  U )   &    |-  M  =  ( normCV `  W )   &    |-  N  =  ( U normOp OLD W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  N  =  ( t  e.  ( Y  ^m  X )  |->  sup ( { x  |  E. z  e.  X  ( ( L `  z )  <_  1  /\  x  =  ( M `  ( t `  z
 ) ) ) } ,  RR* ,  <  )
 ) )
 
Theoremnmooval 22217* The operator norm function. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  L  =  ( normCV `  U )   &    |-  M  =  ( normCV `  W )   &    |-  N  =  ( U normOp OLD W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X --> Y )  ->  ( N `  T )  =  sup ( { x  |  E. z  e.  X  ( ( L `
  z )  <_ 
 1  /\  x  =  ( M `  ( T `
  z ) ) ) } ,  RR* ,  <  ) )
 
Theoremnmosetre 22218* The set in the supremum of the operator norm definition df-nmoo 22199 is a set of reals. (Contributed by NM, 13-Nov-2007.) (New usage is discouraged.)
 |-  Y  =  ( BaseSet `  W )   &    |-  N  =  (
 normCV `  W )   =>    |-  ( ( W  e.  NrmCVec  /\  T : X --> Y ) 
 ->  { x  |  E. z  e.  X  (
 ( M `  z
 )  <_  1  /\  x  =  ( N `  ( T `  z
 ) ) ) }  C_ 
 RR )
 
Theoremnmosetn0 22219* The set in the supremum of the operator norm definition df-nmoo 22199 is nonempty. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   &    |-  M  =  ( normCV `  U )   =>    |-  ( U  e.  NrmCVec  ->  ( N `  ( T `  Z ) )  e.  { x  |  E. y  e.  X  ( ( M `  y )  <_  1  /\  x  =  ( N `  ( T `  y
 ) ) ) }
 )
 
Theoremnmoxr 22220 The norm of an operator is an extended real. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  N  =  ( U normOp OLD W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X --> Y )  ->  ( N `  T )  e.  RR* )
 
Theoremnmooge0 22221 The norm of an operator is nonnegative. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  N  =  ( U normOp OLD W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X --> Y )  -> 
 0  <_  ( N `  T ) )
 
Theoremnmorepnf 22222 The norm of an operator is either real or plus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  N  =  ( U normOp OLD W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X --> Y )  ->  ( ( N `  T )  e.  RR  <->  ( N `  T )  =/=  +oo ) )
 
Theoremnmoreltpnf 22223 The norm of any operator is real iff it is less than plus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  N  =  ( U normOp OLD W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X --> Y )  ->  ( ( N `  T )  e.  RR  <->  ( N `  T )  <  +oo ) )
 
Theoremnmogtmnf 22224 The norm of an operator is greater than minus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  N  =  ( U normOp OLD W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X --> Y )  ->  -oo  <  ( N `  T ) )
 
Theoremnmoolb 22225 A lower bound for an operator norm. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  L  =  ( normCV `  U )   &    |-  M  =  ( normCV `  W )   &    |-  N  =  ( U normOp OLD W )   =>    |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X --> Y ) 
 /\  ( A  e.  X  /\  ( L `  A )  <_  1 ) )  ->  ( M `  ( T `  A ) )  <_  ( N `
  T ) )
 
Theoremnmoubi 22226* An upper bound for an operator norm. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  L  =  ( normCV `  U )   &    |-  M  =  ( normCV `  W )   &    |-  N  =  ( U normOp OLD W )   &    |-  U  e.  NrmCVec   &    |-  W  e.  NrmCVec   =>    |-  ( ( T : X
 --> Y  /\  A  e.  RR* )  ->  ( ( N `  T )  <_  A 
 <-> 
 A. x  e.  X  ( ( L `  x )  <_  1  ->  ( M `  ( T `
  x ) ) 
 <_  A ) ) )
 
Theoremnmoub3i 22227* An upper bound for an operator norm. (Contributed by NM, 12-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  L  =  ( normCV `  U )   &    |-  M  =  ( normCV `  W )   &    |-  N  =  ( U normOp OLD W )   &    |-  U  e.  NrmCVec   &    |-  W  e.  NrmCVec   =>    |-  ( ( T : X
 --> Y  /\  A  e.  RR  /\  A. x  e.  X  ( M `  ( T `  x ) )  <_  ( A  x.  ( L `  x ) ) )  ->  ( N `  T ) 
 <_  ( abs `  A ) )
 
Theoremnmoub2i 22228* An upper bound for an operator norm. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  L  =  ( normCV `  U )   &    |-  M  =  ( normCV `  W )   &    |-  N  =  ( U normOp OLD W )   &    |-  U  e.  NrmCVec   &    |-  W  e.  NrmCVec   =>    |-  ( ( T : X
 --> Y  /\  ( A  e.  RR  /\  0  <_  A )  /\  A. x  e.  X  ( M `  ( T `  x ) )  <_  ( A  x.  ( L `  x ) ) )  ->  ( N `  T )  <_  A )
 
Theoremnmobndi 22229* Two ways to express that an operator is bounded. (Contributed by NM, 11-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  L  =  ( normCV `  U )   &    |-  M  =  ( normCV `  W )   &    |-  N  =  ( U normOp OLD W )   &    |-  U  e.  NrmCVec   &    |-  W  e.  NrmCVec   =>    |-  ( T : X --> Y  ->  ( ( N `
  T )  e. 
 RR 
 <-> 
 E. r  e.  RR  A. y  e.  X  ( ( L `  y
 )  <_  1  ->  ( M `  ( T `
  y ) ) 
 <_  r ) ) )
 
Theoremnmounbi 22230* Two ways two express that an operator is unbounded. (Contributed by NM, 11-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  L  =  ( normCV `  U )   &    |-  M  =  ( normCV `  W )   &    |-  N  =  ( U normOp OLD W )   &    |-  U  e.  NrmCVec   &    |-  W  e.  NrmCVec   =>    |-  ( T : X --> Y  ->  ( ( N `
  T )  = 
 +oo 
 <-> 
 A. r  e.  RR  E. y  e.  X  ( ( L `  y
 )  <_  1  /\  r  <  ( M `  ( T `  y ) ) ) ) )
 
Theoremnmounbseqi 22231* An unbounded operator determines an unbounded sequence. (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 7-Apr-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  L  =  ( normCV `  U )   &    |-  M  =  ( normCV `  W )   &    |-  N  =  ( U normOp OLD W )   &    |-  U  e.  NrmCVec   &    |-  W  e.  NrmCVec   =>    |-  ( ( T : X
 --> Y  /\  ( N `
  T )  = 
 +oo )  ->  E. f
 ( f : NN --> X  /\  A. k  e. 
 NN  ( ( L `
  ( f `  k ) )  <_ 
 1  /\  k  <  ( M `  ( T `
  ( f `  k ) ) ) ) ) )
 
TheoremnmounbseqiOLD 22232* An unbounded operator determines an unbounded sequence. (Contributed by NM, 11-Jan-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  L  =  ( normCV `  U )   &    |-  M  =  ( normCV `  W )   &    |-  N  =  ( U normOp OLD W )   &    |-  U  e.  NrmCVec   &    |-  W  e.  NrmCVec   =>    |-  ( ( T : X
 --> Y  /\  ( N `
  T )  = 
 +oo )  ->  E. f
 ( f : NN --> X  /\  A. k  e. 
 NN  ( ( L `
  ( f `  k ) )  <_ 
 1  /\  k  <  ( M `  ( T `
  ( f `  k ) ) ) ) ) )
 
Theoremnmobndseqi 22233* A bounded sequence determines a bounded operator. (Contributed by NM, 18-Jan-2008.) (Revised by Mario Carneiro, 7-Apr-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  L  =  ( normCV `  U )   &    |-  M  =  ( normCV `  W )   &    |-  N  =  ( U normOp OLD W )   &    |-  U  e.  NrmCVec   &    |-  W  e.  NrmCVec   =>    |-  ( ( T : X
 --> Y  /\  A. f
 ( ( f : NN --> X  /\  A. k  e.  NN  ( L `  ( f `  k ) )  <_ 
 1 )  ->  E. k  e.  NN  ( M `  ( T `  ( f `
  k ) ) )  <_  k )
 )  ->  ( N `  T )  e.  RR )
 
TheoremnmobndseqiOLD 22234* A bounded sequence determines a bounded operator. (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  L  =  ( normCV `  U )   &    |-  M  =  ( normCV `  W )   &    |-  N  =  ( U normOp OLD W )   &    |-  U  e.  NrmCVec   &    |-  W  e.  NrmCVec   =>    |-  ( ( T : X
 --> Y  /\  A. f
 ( ( f : NN --> X  /\  A. k  e.  NN  ( L `  ( f `  k ) )  <_ 
 1 )  ->  E. k  e.  NN  ( M `  ( T `  ( f `
  k ) ) )  <_  k )
 )  ->  ( N `  T )  e.  RR )
 
Theorembloval 22235* The class of bounded linear operators between two normed complex vector spaces. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  N  =  ( U
 normOp OLD W )   &    |-  L  =  ( U  LnOp  W )   &    |-  B  =  ( U  BLnOp  W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  B  =  { t  e.  L  |  ( N `
  t )  <  +oo } )
 
Theoremisblo 22236 The predicate "is a bounded linear operator." (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.)
 |-  N  =  ( U
 normOp OLD W )   &    |-  L  =  ( U  LnOp  W )   &    |-  B  =  ( U  BLnOp  W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( T  e.  B  <->  ( T  e.  L  /\  ( N `  T )  <  +oo ) ) )
 
Theoremisblo2 22237 The predicate "is a bounded linear operator." (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
 |-  N  =  ( U
 normOp OLD W )   &    |-  L  =  ( U  LnOp  W )   &    |-  B  =  ( U  BLnOp  W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( T  e.  B  <->  ( T  e.  L  /\  ( N `  T )  e.  RR ) ) )
 
Theorembloln 22238 A bounded operator is a linear operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
 |-  L  =  ( U 
 LnOp  W )   &    |-  B  =  ( U  BLnOp  W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  B ) 
 ->  T  e.  L )
 
Theoremblof 22239 A bounded operator is an operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  B  =  ( U  BLnOp  W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  B )  ->  T : X --> Y )
 
Theoremnmblore 22240 The norm of a bounded operator is a real number. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  N  =  ( U normOp OLD W )   &    |-  B  =  ( U 
 BLnOp  W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  B )  ->  ( N `  T )  e. 
 RR )
 
Theorem0ofval 22241 The zero operator between two normed complex vector spaces. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  W )   &    |-  O  =  ( U  0op  W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  O  =  ( X  X.  { Z } )
 )
 
Theorem0oval 22242 Value of the zero operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  W )   &    |-  O  =  ( U  0op  W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  A  e.  X )  ->  ( O `  A )  =  Z )
 
Theorem0oo 22243 The zero operator is an operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  Z  =  ( U  0op  W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  Z : X --> Y )
 
Theorem0lno 22244 The zero operator is linear. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
 |-  Z  =  ( U 
 0op  W )   &    |-  L  =  ( U  LnOp  W )   =>    |-  (
 ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  Z  e.  L )
 
Theoremnmoo0 22245 The operator norm of the zero operator. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.)
 |-  N  =  ( U
 normOp OLD W )   &    |-  Z  =  ( U  0op  W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( N `  Z )  =  0 )
 
Theorem0blo 22246 The zero operator is a bounded linear operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
 |-  Z  =  ( U 
 0op  W )   &    |-  B  =  ( U  BLnOp  W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  Z  e.  B )
 
Theoremnmlno0lem 22247 Lemma for nmlno0i 22248. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
 |-  N  =  ( U
 normOp OLD W )   &    |-  Z  =  ( U  0op  W )   &    |-  L  =  ( U 
 LnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e. 
 NrmCVec   &    |-  T  e.  L   &    |-  X  =  (
 BaseSet `  U )   &    |-  Y  =  ( BaseSet `  W )   &    |-  R  =  ( .s OLD `  U )   &    |-  S  =  ( .s
 OLD `  W )   &    |-  P  =  ( 0vec `  U )   &    |-  Q  =  ( 0vec `  W )   &    |-  K  =  ( normCV `  U )   &    |-  M  =  ( normCV `  W )   =>    |-  ( ( N `
  T )  =  0  <->  T  =  Z )
 
Theoremnmlno0i 22248 The norm of a linear operator is zero iff the operator is zero. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.)
 |-  N  =  ( U
 normOp OLD W )   &    |-  Z  =  ( U  0op  W )   &    |-  L  =  ( U 
 LnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e. 
 NrmCVec   =>    |-  ( T  e.  L  ->  ( ( N `  T )  =  0  <->  T  =  Z ) )
 
Theoremnmlno0 22249 The norm of a linear operator is zero iff the operator is zero. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
 |-  N  =  ( U
 normOp OLD W )   &    |-  Z  =  ( U  0op  W )   &    |-  L  =  ( U 
 LnOp  W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  (
 ( N `  T )  =  0  <->  T  =  Z ) )
 
Theoremnmlnoubi 22250* An upper bound for the operator norm of a linear operator, using only the properties of nonzero arguments. (Contributed by NM, 1-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   &    |-  K  =  ( normCV `  U )   &    |-  M  =  ( normCV `  W )   &    |-  N  =  ( U normOp OLD W )   &    |-  L  =  ( U 
 LnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e. 
 NrmCVec   =>    |-  ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A )  /\  A. x  e.  X  ( x  =/= 
 Z  ->  ( M `  ( T `  x ) )  <_  ( A  x.  ( K `  x ) ) ) )  ->  ( N `  T )  <_  A )
 
Theoremnmlnogt0 22251 The norm of a nonzero linear operator is positive. (Contributed by NM, 10-Dec-2007.) (New usage is discouraged.)
 |-  N  =  ( U
 normOp OLD W )   &    |-  Z  =  ( U  0op  W )   &    |-  L  =  ( U 
 LnOp  W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T  =/=  Z  <->  0  <  ( N `  T ) ) )
 
Theoremlnon0 22252* The domain of a nonzero linear operator contains a nonzero vector. (Contributed by NM, 15-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   &    |-  O  =  ( U  0op  W )   &    |-  L  =  ( U 
 LnOp  W )   =>    |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L ) 
 /\  T  =/=  O )  ->  E. x  e.  X  x  =/=  Z )
 
Theoremnmblolbii 22253 A lower bound for the norm of a bounded linear operator. (Contributed by NM, 7-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  L  =  (
 normCV `  U )   &    |-  M  =  (
 normCV `  W )   &    |-  N  =  ( U normOp OLD W )   &    |-  B  =  ( U  BLnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e.  NrmCVec   &    |-  T  e.  B   =>    |-  ( A  e.  X  ->  ( M `  ( T `  A ) )  <_  ( ( N `  T )  x.  ( L `  A ) ) )
 
Theoremnmblolbi 22254 A lower bound for the norm of a bounded linear operator. (Contributed by NM, 10-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  L  =  (
 normCV `  U )   &    |-  M  =  (
 normCV `  W )   &    |-  N  =  ( U normOp OLD W )   &    |-  B  =  ( U  BLnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e.  NrmCVec   =>    |-  ( ( T  e.  B  /\  A  e.  X )  ->  ( M `  ( T `  A ) )  <_  ( ( N `  T )  x.  ( L `  A ) ) )
 
Theoremisblo3i 22255* The predicate "is a bounded linear operator." Definition 2.7-1 of [Kreyszig] p. 91. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  (
 normCV `  U )   &    |-  N  =  (
 normCV `  W )   &    |-  L  =  ( U  LnOp  W )   &    |-  B  =  ( U  BLnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e.  NrmCVec   =>    |-  ( T  e.  B  <->  ( T  e.  L  /\  E. x  e.  RR  A. y  e.  X  ( N `  ( T `  y ) )  <_  ( x  x.  ( M `  y ) ) ) )
 
Theoremblo3i 22256* Properties that determine a bounded linear operator. (Contributed by NM, 13-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  M  =  (
 normCV `  U )   &    |-  N  =  (
 normCV `  W )   &    |-  L  =  ( U  LnOp  W )   &    |-  B  =  ( U  BLnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e.  NrmCVec   =>    |-  ( ( T  e.  L  /\  A  e.  RR  /\ 
 A. y  e.  X  ( N `  ( T `
  y ) ) 
 <_  ( A  x.  ( M `  y ) ) )  ->  T  e.  B )
 
Theoremblometi 22257 Upper bound for the distance between the values of a bounded linear operator. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  C  =  ( IndMet `  U )   &    |-  D  =  ( IndMet `  W )   &    |-  N  =  ( U normOp OLD W )   &    |-  B  =  ( U 
 BLnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e. 
 NrmCVec   =>    |-  ( ( T  e.  B  /\  P  e.  X  /\  Q  e.  X ) 
 ->  ( ( T `  P ) D ( T `  Q ) )  <_  ( ( N `  T )  x.  ( P C Q ) ) )
 
Theoremblocnilem 22258 Lemma for blocni 22259 and lnocni 22260. If a linear operator is continuous at any point, it is bounded. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)
 |-  C  =  ( IndMet `  U )   &    |-  D  =  (
 IndMet `  W )   &    |-  J  =  ( MetOpen `  C )   &    |-  K  =  ( MetOpen `  D )   &    |-  L  =  ( U  LnOp  W )   &    |-  B  =  ( U  BLnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e. 
 NrmCVec   &    |-  T  e.  L   &    |-  X  =  (
 BaseSet `  U )   =>    |-  ( ( P  e.  X  /\  T  e.  ( ( J  CnP  K ) `  P ) )  ->  T  e.  B )
 
Theoremblocni 22259 A linear operator is continuous iff it is bounded. Theorem 2.7-9(a) of [Kreyszig] p. 97. (Contributed by NM, 18-Dec-2007.) (Revised by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)
 |-  C  =  ( IndMet `  U )   &    |-  D  =  (
 IndMet `  W )   &    |-  J  =  ( MetOpen `  C )   &    |-  K  =  ( MetOpen `  D )   &    |-  L  =  ( U  LnOp  W )   &    |-  B  =  ( U  BLnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e. 
 NrmCVec   &    |-  T  e.  L   =>    |-  ( T  e.  ( J  Cn  K )  <->  T  e.  B )
 
Theoremlnocni 22260 If a linear operator is continuous at any point, it is continuous everywhere. Theorem 2.7-9(b) of [Kreyszig] p. 97. (Contributed by NM, 18-Dec-2007.) (New usage is discouraged.)
 |-  C  =  ( IndMet `  U )   &    |-  D  =  (
 IndMet `  W )   &    |-  J  =  ( MetOpen `  C )   &    |-  K  =  ( MetOpen `  D )   &    |-  L  =  ( U  LnOp  W )   &    |-  B  =  ( U  BLnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e. 
 NrmCVec   &    |-  T  e.  L   &    |-  X  =  (
 BaseSet `  U )   =>    |-  ( ( P  e.  X  /\  T  e.  ( ( J  CnP  K ) `  P ) )  ->  T  e.  ( J  Cn  K ) )
 
Theoremblocn 22261 A linear operator is continuous iff it is bounded. Theorem 2.7-9(a) of [Kreyszig] p. 97. (Contributed by NM, 25-Dec-2007.) (New usage is discouraged.)
 |-  C  =  ( IndMet `  U )   &    |-  D  =  (
 IndMet `  W )   &    |-  J  =  ( MetOpen `  C )   &    |-  K  =  ( MetOpen `  D )   &    |-  B  =  ( U  BLnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e.  NrmCVec   &    |-  L  =  ( U 
 LnOp  W )   =>    |-  ( T  e.  L  ->  ( T  e.  ( J  Cn  K )  <->  T  e.  B ) )
 
Theoremblocn2 22262 A bounded linear operator is continuous. (Contributed by NM, 25-Dec-2007.) (New usage is discouraged.)
 |-  C  =  ( IndMet `  U )   &    |-  D  =  (
 IndMet `  W )   &    |-  J  =  ( MetOpen `  C )   &    |-  K  =  ( MetOpen `  D )   &    |-  B  =  ( U  BLnOp  W )   &    |-  U  e.  NrmCVec   &    |-  W  e.  NrmCVec   =>    |-  ( T  e.  B  ->  T  e.  ( J  Cn  K ) )
 
Theoremajfval 22263* The adjoint function. (Contributed by NM, 25-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  P  =  ( .i OLD `  U )   &    |-  Q  =  ( .i
 OLD `  W )   &    |-  A  =  ( U adj W )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  A  =  { <. t ,  s >.  |  (
 t : X --> Y  /\  s : Y --> X  /\  A. x  e.  X  A. y  e.  Y  (
 ( t `  x ) Q y )  =  ( x P ( s `  y ) ) ) } )
 
Theoremhmoval 22264* The set of Hermitian (self-adjoint) operators on a normed complex vector space. (Contributed by NM, 26-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  H  =  ( HmOp `  U )   &    |-  A  =  ( U adj U )   =>    |-  ( U  e.  NrmCVec  ->  H  =  { t  e.  dom  A  |  ( A `  t )  =  t } )
 
Theoremishmo 22265 The predicate "is a hermitian operator." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
 |-  H  =  ( HmOp `  U )   &    |-  A  =  ( U adj U )   =>    |-  ( U  e.  NrmCVec  ->  ( T  e.  H  <->  ( T  e.  dom 
 A  /\  ( A `  T )  =  T ) ) )
 
17.4  Inner product (pre-Hilbert) spaces
 
17.4.1  Definition and basic properties
 
Syntaxccphlo 22266 Extend class notation with the class of all complex inner product spaces (also called pre-Hilbert spaces).
 class  CPreHil OLD
 
Definitiondf-ph 22267* Define the class of all complex inner product spaces. An inner product space is a normed vector space whose norm satisfies the parallelogram law (a property that induces an inner product). Based on Exercise 4(b) of [ReedSimon] p. 63. The vector operation is  g, the scalar product is  s, and the norm is  n. An inner product space is also called a pre-Hilbert space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
 |-  CPreHil
 OLD  =  ( NrmCVec  i^i  { <. <. g ,  s >. ,  n >.  |  A. x  e.  ran  g A. y  e.  ran  g ( ( ( n `  ( x g y ) ) ^ 2 )  +  ( ( n `
  ( x g ( -u 1 s y ) ) ) ^
 2 ) )  =  ( 2  x.  (
 ( ( n `  x ) ^ 2
 )  +  ( ( n `  y ) ^ 2 ) ) ) } )
 
Theoremphnv 22268 Every complex inner product space is a normed complex vector space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
 |-  ( U  e.  CPreHil OLD 
 ->  U  e.  NrmCVec )
 
Theoremphrel 22269 The class of all complex inner product spaces is a relation. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
 |- 
 Rel  CPreHil OLD
 
Theoremphnvi 22270 Every complex inner product space is a normed complex vector space. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
 |-  U  e.  CPreHil OLD   =>    |-  U  e.  NrmCVec
 
Theoremisphg 22271* The predicate "is a complex inner product space." An inner product space is a normed vector space whose norm satisfies the parallelogram law. The vector (group) addition operation is  G, the scalar product is  S, and the norm is  N. An inner product space is also called a pre-Hilbert space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( G  e.  A  /\  S  e.  B  /\  N  e.  C )  ->  ( <. <. G ,  S >. ,  N >.  e.  CPreHil OLD  <->  (
 <. <. G ,  S >. ,  N >.  e.  NrmCVec  /\  A. x  e.  X  A. y  e.  X  (
 ( ( N `  ( x G y ) ) ^ 2 )  +  ( ( N `
  ( x G ( -u 1 S y ) ) ) ^
 2 ) )  =  ( 2  x.  (
 ( ( N `  x ) ^ 2
 )  +  ( ( N `  y ) ^ 2 ) ) ) ) ) )
 
Theoremphop 22272 A complex inner product space in terms of ordered pair components. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
 |-  G  =  ( +v
 `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( U  e.  CPreHil OLD 
 ->  U  =  <. <. G ,  S >. ,  N >. )
 
17.4.2  Examples of pre-Hilbert spaces
 
Theoremcncph 22273 The set of complex numbers is an inner product (pre-Hilbert) space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (Revised by Mario Carneiro, 7-Nov-2013.) (New usage is discouraged.)
 |-  U  =  <. <.  +  ,  x.  >. ,  abs >.   =>    |-  U  e.  CPreHil OLD
 
Theoremelimph 22274 Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   &    |-  U  e. 
 CPreHil OLD   =>    |- 
 if ( A  e.  X ,  A ,  Z )  e.  X
 
Theoremelimphu 22275 Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem. (Contributed by NM, 6-May-2007.) (New usage is discouraged.)
 |- 
 if ( U  e.  CPreHil OLD
 ,  U ,  <. <.  +  ,  x.  >. ,  abs >.
 )  e.  CPreHil OLD
 
17.4.3  Properties of pre-Hilbert spaces
 
Theoremisph 22276* The predicate "is an inner product space." (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  (
 normCV `  U )   =>    |-  ( U  e.  CPreHil OLD  <->  ( U  e.  NrmCVec  /\  A. x  e.  X  A. y  e.  X  ( ( ( N `  ( x G y ) ) ^ 2 )  +  ( ( N `  ( x M y ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `  x ) ^ 2 )  +  ( ( N `  y ) ^ 2
 ) ) ) ) )
 
Theoremphpar2 22277 The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  M  =  ( -v `  U )   &    |-  N  =  (
 normCV `  U )   =>    |-  ( ( U  e.  CPreHil OLD  /\  A  e.  X  /\  B  e.  X )  ->  ( ( ( N `
  ( A G B ) ) ^
 2 )  +  (
 ( N `  ( A M B ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `
  A ) ^
 2 )  +  (
 ( N `  B ) ^ 2 ) ) ) )
 
Theoremphpar 22278 The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   =>    |-  ( ( U  e.  CPreHil OLD  /\  A  e.  X  /\  B  e.  X )  ->  ( ( ( N `  ( A G B ) ) ^ 2 )  +  ( ( N `  ( A G ( -u 1 S B ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `  A ) ^ 2 )  +  ( ( N `  B ) ^ 2
 ) ) ) )
 
Theoremip0i 22279 A slight variant of Equation 6.46 of [Ponnusamy] p. 362, where  J is either 1 or -1 to represent +-1. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   &    |-  C  e.  X   &    |-  N  =  ( normCV `  U )   &    |-  J  e.  CC   =>    |-  ( ( ( ( N `  ( ( A G B ) G ( J S C ) ) ) ^ 2 )  -  ( ( N `  ( ( A G B ) G (
 -u J S C ) ) ) ^
 2 ) )  +  ( ( ( N `
  ( ( A G ( -u 1 S B ) ) G ( J S C ) ) ) ^
 2 )  -  (
 ( N `  (
 ( A G (
 -u 1 S B ) ) G (
 -u J S C ) ) ) ^
 2 ) ) )  =  ( 2  x.  ( ( ( N `
  ( A G ( J S C ) ) ) ^ 2
 )  -  ( ( N `  ( A G ( -u J S C ) ) ) ^ 2 ) ) )
 
Theoremip1ilem 22280 Lemma for ip1i 22281. (Contributed by NM, 21-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   &    |-  C  e.  X   &    |-  N  =  ( normCV `  U )   &    |-  J  e.  CC   =>    |-  ( ( ( A G B ) P C )  +  (
 ( A G (
 -u 1 S B ) ) P C ) )  =  (
 2  x.  ( A P C ) )
 
Theoremip1i 22281 Equation 6.47 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   &    |-  C  e.  X   =>    |-  (
 ( ( A G B ) P C )  +  ( ( A G ( -u 1 S B ) ) P C ) )  =  ( 2  x.  ( A P C ) )
 
Theoremip2i 22282 Equation 6.48 of [Ponnusamy] p. 362. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   =>    |-  ( ( 2 S A ) P B )  =  ( 2  x.  ( A P B ) )
 
Theoremipdirilem 22283 Lemma for ipdiri 22284. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   &    |-  C  e.  X   =>    |-  (
 ( A G B ) P C )  =  ( ( A P C )  +  ( B P C ) )
 
Theoremipdiri 22284 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   =>    |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) 
 ->  ( ( A G B ) P C )  =  ( ( A P C )  +  ( B P C ) ) )
 
Theoremipasslem1 22285 Lemma for ipassi 22295. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  B  e.  X   =>    |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) )
 
Theoremipasslem2 22286 Lemma for ipassi 22295. Show the inner product associative law for nonpositive integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  B  e.  X   =>    |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( (
 -u N S A ) P B )  =  ( -u N  x.  ( A P B ) ) )
 
Theoremipasslem3 22287 Lemma for ipassi 22295. Show the inner product associative law for all integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  B  e.  X   =>    |-  ( ( N  e.  ZZ  /\  A  e.  X )  ->  (
 ( N S A ) P B )  =  ( N  x.  ( A P B ) ) )
 
Theoremipasslem4 22288 Lemma for ipassi 22295. Show the inner product associative law for positive integer reciprocals. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  B  e.  X   =>    |-  ( ( N  e.  NN  /\  A  e.  X )  ->  (
 ( ( 1  /  N ) S A ) P B )  =  ( ( 1  /  N )  x.  ( A P B ) ) )
 
Theoremipasslem5 22289 Lemma for ipassi 22295. Show the inner product associative law for rational numbers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  B  e.  X   =>    |-  ( ( C  e.  QQ  /\  A  e.  X )  ->  (
 ( C S A ) P B )  =  ( C  x.  ( A P B ) ) )
 
Theoremipasslem7 22290* Lemma for ipassi 22295. Show that  ( ( w S A ) P B )  -  (
w  x.  ( A P B ) ) is continuous on  RR. (Contributed by NM, 23-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   &    |-  F  =  ( w  e.  RR  |->  ( ( ( w S A ) P B )  -  ( w  x.  ( A P B ) ) ) )   &    |-  J  =  ( topGen `  ran  (,) )   &    |-  K  =  ( TopOpen ` fld )   =>    |-  F  e.  ( J  Cn  K )
 
Theoremipasslem8 22291* Lemma for ipassi 22295. By ipasslem5 22289, 
F is 0 for all  QQ; since it is continuous and 
QQ is dense in  RR by qdensere2 18781, we conclude  F is 0 for all  RR. (Contributed by NM, 24-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   &    |-  F  =  ( w  e.  RR  |->  ( ( ( w S A ) P B )  -  ( w  x.  ( A P B ) ) ) )   =>    |-  F : RR --> { 0 }
 
Theoremipasslem9 22292 Lemma for ipassi 22295. Conclude from ipasslem8 22291 the inner product associative law for real numbers. (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   =>    |-  ( C  e.  RR  ->  ( ( C S A ) P B )  =  ( C  x.  ( A P B ) ) )
 
Theoremipasslem10 22293 Lemma for ipassi 22295. Show the inner product associative law for the imaginary number  _i. (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   &    |-  N  =  (
 normCV `  U )   =>    |-  ( ( _i S A ) P B )  =  ( _i  x.  ( A P B ) )
 
Theoremipasslem11 22294 Lemma for ipassi 22295. Show the inner product associative law for all complex numbers. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   =>    |-  ( C  e.  CC  ->  ( ( C S A ) P B )  =  ( C  x.  ( A P B ) ) )
 
Theoremipassi 22295 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e.  CPreHil OLD   =>    |-  ( ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) 
 ->  ( ( A S B ) P C )  =  ( A  x.  ( B P C ) ) )
 
Theoremdipdir 22296 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  CPreHil OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( ( A G B ) P C )  =  ( ( A P C )  +  ( B P C ) ) )
 
Theoremdipdi 22297 Distributive law for inner product. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  CPreHil OLD  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( A P ( B G C ) )  =  ( ( A P B )  +  ( A P C ) ) )
 
Theoremip2dii 22298 Inner product of two sums. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  U  e. 
 CPreHil OLD   &    |-  A  e.  X   &    |-  B  e.  X   &    |-  C  e.  X   &    |-  D  e.  X   =>    |-  ( ( A G B ) P ( C G D ) )  =  ( ( ( A P C )  +  ( B P D ) )  +  ( ( A P D )  +  ( B P C ) ) )
 
Theoremdipass 22299 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i
 OLD `  U )   =>    |-  (
 ( U  e.  CPreHil OLD  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A S B ) P C )  =  ( A  x.  ( B P C ) ) )
 
Theoremdipassr 22300 "Associative" law for second argument of inner product (compare dipass 22299). (Contributed by NM, 22-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  P  =  ( .i
 OLD `  U )   =>    |-  (
 ( U  e.  CPreHil OLD  /\  ( A  e.  X  /\  B  e.  CC  /\  C  e.  X )
 )  ->  ( A P ( B S C ) )  =  ( ( * `  B )  x.  ( A P C ) ) )
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