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Theorem List for Metamath Proof Explorer - 21101-21200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsmcn 21101 Scalar multiplication is jointly continuous in both arguments. (Contributed by NM, 16-Jun-2009.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.)
 |-  C  =  ( IndMet `  U )   &    |-  J  =  (
 MetOpen `  C )   &    |-  S  =  ( .s OLD `  U )   &    |-  K  =  ( TopOpen ` fld )   =>    |-  ( U  e.  NrmCVec  ->  S  e.  ( ( K  tX  J )  Cn  J ) )
 
Theoremvmcn 21102 Vector subtraction is jointly continuous in both arguments. (Contributed by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)
 |-  C  =  ( IndMet `  U )   &    |-  J  =  (
 MetOpen `  C )   &    |-  M  =  ( -v `  U )   =>    |-  ( U  e.  NrmCVec  ->  M  e.  ( ( J  tX  J )  Cn  J ) )
 
15.4.4  Inner product
 
Syntaxcdip 21103 Extend class notation with the class inner product functions.
 class  .i OLD
 
Definitiondf-dip 21104* Define a function that maps a complex normed vector space to its inner product operation in case its norm satisfies the parallelogram identity (otherwise the operation is still defined, but not meaningful). Based on Exercise 4(a) of [ReedSimon] p. 63 and Theorem 6.44 of [Ponnusamy] p. 361. Vector addition is  ( 1st `  w
), the scalar product is  ( 2nd `  w
), and the norm is  n. (Contributed by NM, 10-Apr-2007.) (New usage is discouraged.)
 |- 
 .i OLD  =  ( u  e.  NrmCVec  |->  ( x  e.  ( BaseSet `  u ) ,  y  e.  ( BaseSet `  u )  |->  ( sum_ k  e.  (
 1 ... 4 ) ( ( _i ^ k
 )  x.  ( ( ( normCV `  u ) `  ( x ( +v `  u ) ( ( _i ^ k ) ( .s OLD `  u ) y ) ) ) ^ 2 ) )  /  4 ) ) )
 
Theoremdipfval 21105* The inner product function on a normed complex vector space. The definition is meaningful for vector spaces that are also inner product spaces, i.e. satisfy the parallelogram law. (Contributed by NM, 10-Apr-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( U  e.  NrmCVec  ->  P  =  ( x  e.  X ,  y  e.  X  |->  ( sum_ k  e.  ( 1 ... 4
 ) ( ( _i
 ^ k )  x.  ( ( N `  ( x G ( ( _i ^ k ) S y ) ) ) ^ 2 ) )  /  4 ) ) )
 
Theoremipval 21106* Value of the inner product. The definition is meaningful for normed complex vector spaces that are also inner product spaces, i.e. satisfy the parallelogram law, although for convenience we define it for any normed complex vector space. The vector (group) addition operation is  G, the scalar product is  S, the norm is  N, and the set of vectors is  X. Equation 6.45 of [Ponnusamy] p. 361. (Contributed by NM, 31-Jan-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  =  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k )  x.  ( ( N `
  ( A G ( ( _i ^
 k ) S B ) ) ) ^
 2 ) )  / 
 4 ) )
 
Theoremipval2lem2 21107 Lemma for ipval3 21112. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  C  e.  CC )  ->  ( ( N `
  ( A G ( C S B ) ) ) ^ 2
 )  e.  RR )
 
Theoremipval2lem3 21108 Lemma for ipval3 21112. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( ( N `  ( A G B ) ) ^ 2 )  e.  RR )
 
Theoremipval2lem4 21109 Lemma for ipval3 21112. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  C  e.  CC )  ->  ( ( N `
  ( A G ( C S B ) ) ) ^ 2
 )  e.  CC )
 
Theoremipval2 21110 Expansion of the inner product value ipval 21106. (Contributed by NM, 31-Jan-2007.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  =  ( (
 ( ( ( N `
  ( A G B ) ) ^
 2 )  -  (
 ( N `  ( A G ( -u 1 S B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( N `
  ( A G ( _i S B ) ) ) ^ 2
 )  -  ( ( N `  ( A G ( -u _i S B ) ) ) ^ 2 ) ) ) )  /  4
 ) )
 
Theorem4ipval2 21111 Four times the inner product value ipval3 21112, useful for simplifying certain proofs. (Contributed by NM, 10-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( 4  x.  ( A P B ) )  =  ( ( ( ( N `  ( A G B ) ) ^ 2 )  -  ( ( N `  ( A G ( -u 1 S B ) ) ) ^ 2 ) )  +  ( _i 
 x.  ( ( ( N `  ( A G ( _i S B ) ) ) ^ 2 )  -  ( ( N `  ( A G ( -u _i S B ) ) ) ^ 2 ) ) ) ) )
 
Theoremipval3 21112 Expansion of the inner product value ipval 21106. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  M  =  ( -v
 `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  =  (
 ( ( ( ( N `  ( A G B ) ) ^ 2 )  -  ( ( N `  ( A M B ) ) ^ 2 ) )  +  ( _i 
 x.  ( ( ( N `  ( A G ( _i S B ) ) ) ^ 2 )  -  ( ( N `  ( A M ( _i S B ) ) ) ^ 2 ) ) ) )  / 
 4 ) )
 
Theoremipval2lem5 21113 Lemma for ipval3 21112. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  M  =  ( -v
 `  U )   =>    |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  C  e.  CC )  ->  ( ( N `  ( A M ( C S B ) ) ) ^ 2 )  e. 
 RR )
 
Theoremipval2lem6 21114 Lemma for ipval3 21112. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  M  =  ( -v
 `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( ( N `
  ( A M B ) ) ^
 2 )  e.  RR )
 
Theorem4ipval3 21115 Four times the inner product value ipval3 21112, useful for simplifying certain proofs. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  G  =  ( +v `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  P  =  ( .i OLD `  U )   &    |-  M  =  ( -v
 `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( 4  x.  ( A P B ) )  =  (
 ( ( ( N `
  ( A G B ) ) ^
 2 )  -  (
 ( N `  ( A M B ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( N `
  ( A G ( _i S B ) ) ) ^ 2
 )  -  ( ( N `  ( A M ( _i S B ) ) ) ^ 2 ) ) ) ) )
 
Theoremipidsq 21116 The inner product of a vector with itself is the square of the vector's norm. Equation I4 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  N  =  (
 normCV `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( A P A )  =  ( ( N `  A ) ^
 2 ) )
 
Theoremipnm 21117 Norm expressed in terms of inner product. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  N  =  (
 normCV `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( N `  A )  =  ( sqr `  ( A P A ) ) )
 
Theoremdipcl 21118 An inner product is a complex number. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
 
Theoremipf 21119 Mapping for the inner product operation. (Contributed by NM, 28-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( U  e.  NrmCVec  ->  P : ( X  X.  X ) --> CC )
 
Theoremdipcj 21120 The complex conjugate of an inner product reverses its arguments. Equation I1 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( * `  ( A P B ) )  =  ( B P A ) )
 
Theoremipipcj 21121 An inner product times its conjugate. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A P B )  x.  ( B P A ) )  =  ( ( abs `  ( A P B ) ) ^ 2
 ) )
 
Theoremdiporthcom 21122 Orthogonality (meaning inner product is 0) is commutative. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A P B )  =  0  <->  ( B P A )  =  0 ) )
 
Theoremdip0r 21123 Inner product with a zero second argument. (Contributed by NM, 5-Feb-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( A P Z )  =  0 )
 
Theoremdip0l 21124 Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. (Contributed by NM, 5-Feb-2007.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( Z P A )  =  0 )
 
Theoremipz 21125 The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of [Kreyszig] p. 129. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Z  =  (
 0vec `  U )   &    |-  P  =  ( .i OLD `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  A  e.  X ) 
 ->  ( ( A P A )  =  0  <->  A  =  Z ) )
 
Theoremdipcn 21126 Inner product is jointly continuous in both arguments. (Contributed by NM, 21-Aug-2007.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
 |-  P  =  ( .i
 OLD `  U )   &    |-  C  =  ( IndMet `  U )   &    |-  J  =  ( MetOpen `  C )   &    |-  K  =  ( TopOpen ` fld )   =>    |-  ( U  e.  NrmCVec  ->  P  e.  ( ( J  tX  J )  Cn  K ) )
 
15.4.5  Subspaces
 
Syntaxcss 21127 Extend class notation with the class of all subspaces of complex normed vector spaces.
 class  SubSp
 
Definitiondf-ssp 21128* Define the class of all subspaces of complex normed vector spaces. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
 |- 
 SubSp  =  ( u  e. 
 NrmCVec 
 |->  { w  e.  NrmCVec  |  ( ( +v `  w )  C_  ( +v
 `  u )  /\  ( .s OLD `  w )  C_  ( .s OLD `  u )  /\  ( normCV `  w )  C_  ( normCV `  u ) ) }
 )
 
Theoremsspval 21129* The set of all subspaces of a normed complex vector space. (Contributed by NM, 26-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  G  =  ( +v
 `  U )   &    |-  S  =  ( .s OLD `  U )   &    |-  N  =  ( normCV `  U )   &    |-  H  =  ( SubSp `  U )   =>    |-  ( U  e.  NrmCVec  ->  H  =  { w  e. 
 NrmCVec  |  ( ( +v
 `  w )  C_  G  /\  ( .s OLD `  w )  C_  S  /\  ( normCV `  w )  C_  N ) } )
 
Theoremisssp 21130 The predicate "is a subspace." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
 |-  G  =  ( +v
 `  U )   &    |-  F  =  ( +v `  W )   &    |-  S  =  ( .s
 OLD `  U )   &    |-  R  =  ( .s OLD `  W )   &    |-  N  =  ( normCV `  U )   &    |-  M  =  ( normCV `  W )   &    |-  H  =  ( SubSp `  U )   =>    |-  ( U  e.  NrmCVec  ->  ( W  e.  H  <->  ( W  e.  NrmCVec  /\  ( F  C_  G  /\  R  C_  S  /\  M  C_  N ) ) ) )
 
Theoremsspid 21131 A normed complex vector space is a subspace of itself. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
 |-  H  =  ( SubSp `  U )   =>    |-  ( U  e.  NrmCVec  ->  U  e.  H )
 
Theoremsspnv 21132 A subspace is a normed complex vector space. (Contributed by NM, 27-Jan-2008.) (New usage is discouraged.)
 |-  H  =  ( SubSp `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  H ) 
 ->  W  e.  NrmCVec )
 
Theoremsspba 21133 The base set of a subspace is included in the parent base set. (Contributed by NM, 27-Jan-2008.) (New usage is discouraged.)
 |-  X  =  ( BaseSet `  U )   &    |-  Y  =  (
 BaseSet `  W )   &    |-  H  =  ( SubSp `  U )   =>    |-  (
 ( U  e.  NrmCVec  /\  W  e.  H )  ->  Y  C_  X )
 
Theoremsspg 21134 Vector addition on a subspace is a restriction of vector addition on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
 |-  Y  =  ( BaseSet `  W )   &    |-  G  =  ( +v `  U )   &    |-  F  =  ( +v `  W )   &    |-  H  =  (
 SubSp `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  F  =  ( G  |`  ( Y  X.  Y ) ) )
 
Theoremsspgval 21135 Vector addition on a subspace in terms of vector addition on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
 |-  Y  =  ( BaseSet `  W )   &    |-  G  =  ( +v `  U )   &    |-  F  =  ( +v `  W )   &    |-  H  =  (
 SubSp `  U )   =>    |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y )
 )  ->  ( A F B )  =  ( A G B ) )
 
Theoremssps 21136 Scalar multiplication on a subspace is a restriction of scalar multiplication on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
 |-  Y  =  ( BaseSet `  W )   &    |-  S  =  ( .s OLD `  U )   &    |-  R  =  ( .s
 OLD `  W )   &    |-  H  =  ( SubSp `  U )   =>    |-  (
 ( U  e.  NrmCVec  /\  W  e.  H )  ->  R  =  ( S  |`  ( CC  X.  Y ) ) )
 
Theoremsspsval 21137 Scalar multiplication on a subspace in terms of scalar multiplication on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
 |-  Y  =  ( BaseSet `  W )   &    |-  S  =  ( .s OLD `  U )   &    |-  R  =  ( .s
 OLD `  W )   &    |-  H  =  ( SubSp `  U )   =>    |-  (
 ( ( U  e.  NrmCVec  /\  W  e.  H ) 
 /\  ( A  e.  CC  /\  B  e.  Y ) )  ->  ( A R B )  =  ( A S B ) )
 
Theoremsspmlem 21138* Lemma for sspm 21140 and others. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
 |-  Y  =  ( BaseSet `  W )   &    |-  H  =  (
 SubSp `  U )   &    |-  (
 ( ( U  e.  NrmCVec  /\  W  e.  H ) 
 /\  ( x  e.  Y  /\  y  e.  Y ) )  ->  ( x F y )  =  ( x G y ) )   &    |-  ( W  e.  NrmCVec  ->  F :
 ( Y  X.  Y )
 --> R )   &    |-  ( U  e.  NrmCVec  ->  G : ( (
 BaseSet `  U )  X.  ( BaseSet `  U )
 ) --> S )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  F  =  ( G  |`  ( Y  X.  Y ) ) )
 
Theoremsspmval 21139 Vector addition on a subspace in terms of vector addition on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
 |-  Y  =  ( BaseSet `  W )   &    |-  M  =  ( -v `  U )   &    |-  L  =  ( -v `  W )   &    |-  H  =  (
 SubSp `  U )   =>    |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y )
 )  ->  ( A L B )  =  ( A M B ) )
 
Theoremsspm 21140 Vector subtraction on a subspace is a restriction of vector subtraction on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
 |-  Y  =  ( BaseSet `  W )   &    |-  M  =  ( -v `  U )   &    |-  L  =  ( -v `  W )   &    |-  H  =  (
 SubSp `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  L  =  ( M  |`  ( Y  X.  Y ) ) )
 
Theoremsspz 21141 The zero vector of a subspace is the same as the parent's. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
 |-  Z  =  ( 0vec `  U )   &    |-  Q  =  (
 0vec `  W )   &    |-  H  =  ( SubSp `  U )   =>    |-  (
 ( U  e.  NrmCVec  /\  W  e.  H )  ->  Q  =  Z )
 
Theoremsspn 21142 The norm on a subspace is a restriction of the norm on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
 |-  Y  =  ( BaseSet `  W )   &    |-  N  =  (
 normCV `  U )   &    |-  M  =  (
 normCV `  W )   &    |-  H  =  (
 SubSp `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  M  =  ( N  |`  Y ) )
 
Theoremsspnval 21143 The norm on a subspace in terms of the norm on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
 |-  Y  =  ( BaseSet `  W )   &    |-  N  =  (
 normCV `  U )   &    |-  M  =  (
 normCV `  W )   &    |-  H  =  (
 SubSp `  U )   =>    |-  ( ( U  e.  NrmCVec  /\  W  e.  H  /\  A  e.  Y )  ->  ( M `  A )  =  ( N `  A ) )
 
Theoremsspival 21144 The inner product on a subspace in terms of the inner product on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
 |-  Y  =  ( BaseSet `  W )   &    |-  P  =  ( .i OLD `  U )   &    |-  Q  =  ( .i
 OLD `  W )   &    |-  H  =  ( SubSp `  U )   =>    |-  (
 ( ( U  e.  NrmCVec  /\  W  e.  H ) 
 /\  ( A  e.  Y  /\  B  e.  Y ) )  ->  ( A Q B )  =  ( A P B ) )
 
Theoremsspi 21145 The inner product on a subspace is a restriction of the inner product on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
 |-  Y  =  ( BaseSet `  W )   &    |-  P  =  ( .i OLD `  U )   &    |-  Q  =  ( .i
 OLD `  W )   &    |-  H  =  ( SubSp `  U )   =>    |-  (
 ( U  e.  NrmCVec  /\  W  e.  H )  ->  Q  =  ( P  |`  ( Y  X.  Y ) ) )
 
Theoremsspimsval 21146 The induced metric on a subspace in terms of the induced metric on the parent space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
 |-  Y  =  ( BaseSet `  W )   &    |-  D  =  (
 IndMet `  U )   &    |-  C  =  ( IndMet `  W )   &    |-  H  =  ( SubSp `  U )   =>    |-  (
 ( ( U  e.  NrmCVec  /\  W  e.  H ) 
 /\  ( A  e.  Y  /\  B  e.  Y ) )  ->  ( A C B )  =  ( A D B ) )
 
Theoremsspims 21147 The induced metric on a subspace is a restriction of the induced metric on the parent space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
 |-  Y  =  ( BaseSet `  W )   &    |-  D  =  (
 IndMet `  U )   &    |-  C  =  ( IndMet `  W )   &    |-  H  =  ( SubSp `  U )   =>    |-  (
 ( U  e.  NrmCVec  /\  W  e.  H )  ->  C  =  ( D  |`  ( Y  X.  Y ) ) )
 
15.5  Operators on complex vector spaces
 
15.5.1  Definitions and basic properties
 
Syntaxclno 21148 Extend class notation with the class of linear operators on normed complex vector spaces.
 class  LnOp
 
Syntaxcnmoo 21149 Extend class notation with the class of operator norms on normed complex vector spaces.
 class  normOp OLD
 
Syntaxcblo 21150 Extend class notation with the class of bounded linear operators on normed complex vector spaces.
 class  BLnOp
 
Syntaxc0o 21151 Extend class notation with the class of zero operators on normed complex vector spaces.
 class  0op
 
Definitiondf-lno 21152* Define the class of linear operators between two normed complex vector spaces. In the literature, an operator may be a partial function, i.e. the domain of an operator is not necessarily the entire vector space. However, since the domain of a linear operator is a vector subspace, we define it with a complete function for convenience and will use subset relations to specify the partial function case. (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.)
 |- 
 LnOp  =  ( u  e. 
 NrmCVec ,  w  e.  NrmCVec  |->  { t  e.  ( (
 BaseSet `  w )  ^m  ( BaseSet `  u )
 )  |  A. x  e.  CC  A. y  e.  ( BaseSet `  u ) A. z  e.  ( BaseSet `  u ) ( t `
  ( ( x ( .s OLD `  u ) y ) ( +v `  u ) z ) )  =  ( ( x ( .s OLD `  w ) ( t `  y ) ) ( +v `  w ) ( t `  z
 ) ) } )
 
Definitiondf-nmoo 21153* Define the norm of an operator between two normed complex vector spaces. This definition produces an operator norm function for each pair of vector spaces 
<. u ,  w >.. Based on definition of linear operator norm in [AkhiezerGlazman] p. 39, although we define it for all operators for convenience. It isn't necessarily meaningful for nonlinear operators, since it doesn't take into account operator values at vectors with norm greater than 1. See Equation 2 of [Kreyszig] p. 92 for a definition that does (although it ignores the value at the zero vector). However, operator norms are rarely if ever used for nonlinear operators. (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.)
 |-  normOp OLD  =  ( u  e.  NrmCVec ,  w  e.  NrmCVec  |->  ( t  e.  (
 ( BaseSet `  w )  ^m  ( BaseSet `  u )
 )  |->  sup ( { x  |  E. z  e.  ( BaseSet `  u ) ( ( ( normCV `  u ) `  z )  <_  1  /\  x  =  ( ( normCV `  w ) `  (
 t `  z )
 ) ) } ,  RR*
 ,  <  ) )
 )
 
Definitiondf-blo 21154* Define the class of bounded linear operators between two normed complex vector spaces. (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.)
 |- 
 BLnOp  =  ( u  e. 
 NrmCVec ,  w  e.  NrmCVec  |->  { t  e.  ( u 
 LnOp  w )  |  ( ( u normOp OLD w ) `  t )  <  +oo } )
 
Definitiondf-0o 21155* 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 21156 Adjoint of an operator.
 class  adj
 
Syntaxchmo 21157 Set of Hermitional (self-adjoint) operators.
 class  HmOp
 
Definitiondf-aj 21158* 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 21159* 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 21160* 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 21161* 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 21162 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 21163 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 21164 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 21165 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 21166 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 21167 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 21168 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 21169 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 21170* 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 21171* 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 21172* The set in the supremum of the operator norm definition df-nmoo 21153 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 21173* The set in the supremum of the operator norm definition df-nmoo 21153 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 21174 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 21175 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 21176 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 21177 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 21178 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 21179 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 21180* 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 21181* 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 21182* 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 21183* 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 21184* 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 21185* 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 21186* An unbounded operator determines an unbounded sequence. (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 )  ->  E. f
 ( f : NN --> X  /\  A. k  e. 
 NN  ( ( L `
  ( f `  k ) )  <_ 
 1  /\  k  <  ( M `  ( T `
  ( f `  k ) ) ) ) ) )
 
Theoremnmobndseqi 21187* 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 21188* A bounded sequence determines a bounded operator. (Contributed by NM, 18-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  /\  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 21189* 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 21190 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 21191 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 21192 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 21193 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 21194 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 21195 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 21196 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 21197 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 21198 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 21199 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 21200 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 )
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