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Theorem List for Metamath Proof Explorer - 21301-21400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsspid 21301 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 21302 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 21303 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 21304 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 21305 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 21306 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 21307 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 21308* Lemma for sspm 21310 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 21309 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 21310 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 21311 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 21312 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 21313 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 21314 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 21315 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 21316 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 21317 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 ) ) )
 
16.3  Operators on complex vector spaces
 
16.3.1  Definitions and basic properties
 
Syntaxclno 21318 Extend class notation with the class of linear operators on normed complex vector spaces.
 class  LnOp
 
Syntaxcnmoo 21319 Extend class notation with the class of operator norms on normed complex vector spaces.
 class  normOp OLD
 
Syntaxcblo 21320 Extend class notation with the class of bounded linear operators on normed complex vector spaces.
 class  BLnOp
 
Syntaxc0o 21321 Extend class notation with the class of zero operators on normed complex vector spaces.
 class  0op
 
Definitiondf-lno 21322* 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 21323* 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 21324* 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 21325* 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 21326 Adjoint of an operator.
 class  adj
 
Syntaxchmo 21327 Set of Hermitional (self-adjoint) operators.
 class  HmOp
 
Definitiondf-aj 21328* 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 21329* 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 21330* 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 21331* 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 21332 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 21333 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 21334 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 21335 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 21336 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 21337 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 21338 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 21339 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 21340* 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 21341* 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 21342* The set in the supremum of the operator norm definition df-nmoo 21323 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 21343* The set in the supremum of the operator norm definition df-nmoo 21323 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 21344 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 21345 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 21346 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 21347 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 21348 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 21349 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 21350* 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 21351* 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 21352* 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 21353* 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 21354* 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 21355* 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 21356* 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 21357* 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 21358* 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 21359* 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 21360 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 21361 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 21362 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 21363 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 21364 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 21365 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 21366 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 21367 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 21368 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 21369 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 21370 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 21371 Lemma for nmlno0i 21372. (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 21372 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 21373 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 21374* 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 21375 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 21376* 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 21377 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 21378 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 21379* 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 21380* 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 21381 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 21382 Lemma for blocni 21383 and lnocni 21384. 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 21383 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 21384 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 21385 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 21386 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 21387* 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 21388* 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 21389 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 ) ) )
 
16.4  Inner product (pre-Hilbert) spaces
 
16.4.1  Definition and basic properties
 
Syntaxccphlo 21390 Extend class notation with the class of all complex inner product spaces (also called pre-Hilbert spaces).
 class  CPreHil OLD
 
Definitiondf-ph 21391* 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 21392 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 21393 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 21394 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 21395* 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 21396 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 >. )
 
16.4.2  Examples of pre-Hilbert spaces
 
Theoremcncph 21397 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 21398 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 21399 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
 
16.4.3  Properties of pre-Hilbert spaces
 
Theoremisph 21400* 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
 ) ) ) ) )
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