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Theorem List for Metamath Proof Explorer - 22201-22300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
17.2.4  Inner product
 
Syntaxcdip 22201 Extend class notation with the class inner product functions.
 class  .i OLD
 
Definitiondf-dip 22202* 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 22203* 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 22204* 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 22205 Lemma for ipval3 22210. (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 22206 Lemma for ipval3 22210. (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 22207 Lemma for ipval3 22210. (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 22208 Expansion of the inner product value ipval 22204. (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 22209 Four times the inner product value ipval3 22210, 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 22210 Expansion of the inner product value ipval 22204. (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 22211 Lemma for ipval3 22210. (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 22212 Lemma for ipval3 22210. (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 22213 Four times the inner product value ipval3 22210, 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 22214 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 22215 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 22216 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 22217 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 22218 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 22219 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 22220 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 22221 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 22222 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 22223 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 22224 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 ) )
 
17.2.5  Subspaces
 
Syntaxcss 22225 Extend class notation with the class of all subspaces of complex normed vector spaces.
 class  SubSp
 
Definitiondf-ssp 22226* 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 22227* 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 22228 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 22229 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 22230 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 22231 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 22232 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 22233 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 22234 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 22235 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 22236* Lemma for sspm 22238 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 22237 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 22238 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 22239 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 22240 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 22241 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 22242 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 22243 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 22244 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 22245 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 ) ) )
 
17.3  Operators on complex vector spaces
 
17.3.1  Definitions and basic properties
 
Syntaxclno 22246 Extend class notation with the class of linear operators on normed complex vector spaces.
 class  LnOp
 
Syntaxcnmoo 22247 Extend class notation with the class of operator norms on normed complex vector spaces.
 class  normOp OLD
 
Syntaxcblo 22248 Extend class notation with the class of bounded linear operators on normed complex vector spaces.
 class  BLnOp
 
Syntaxc0o 22249 Extend class notation with the class of zero operators on normed complex vector spaces.
 class  0op
 
Definitiondf-lno 22250* 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 22251* 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 22252* 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 22253* 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 22254 Adjoint of an operator.
 class  adj
 
Syntaxchmo 22255 Set of Hermitional (self-adjoint) operators.
 class  HmOp
 
Definitiondf-aj 22256* 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 22257* 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 22258* 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 22259* 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 22260 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 22261 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 22262 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 22263 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 22264 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 22265 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 22266 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 22267 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 22268* 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 22269* 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 22270* The set in the supremum of the operator norm definition df-nmoo 22251 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 22271* The set in the supremum of the operator norm definition df-nmoo 22251 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 22272 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 22273 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 22274 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 22275 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 22276 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 22277 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 22278* 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 22279* 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 22280* 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 22281* 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 22282* 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 22283* 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 22284* 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 22285* 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 22286* 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 22287* 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 22288 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 22289 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 22290 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 22291 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 22292 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 22293 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 22294 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 22295 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 22296 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 22297 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 22298 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 22299 Lemma for nmlno0i 22300. (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 22300 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 ) )
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