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Theorem List for Metamath Proof Explorer - 18001-18100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtngngpd 18001* Derive the axioms for a normed group from the axioms for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  T  =  ( G toNrmGrp  N )   &    |-  X  =  (
 Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  N : X --> RR )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  ( ( N `  x )  =  0  <->  x  =  .0.  ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X )
 )  ->  ( N `  ( x  .-  y
 ) )  <_  (
 ( N `  x )  +  ( N `  y ) ) )   =>    |-  ( ph  ->  T  e. NrmGrp )
 
Theoremtngngp 18002* Derive the axioms for a normed group from the axioms for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  T  =  ( G toNrmGrp  N )   &    |-  X  =  (
 Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( N : X --> RR  ->  ( T  e. NrmGrp  <->  ( G  e.  Grp  /\  A. x  e.  X  (
 ( ( N `  x )  =  0  <->  x  =  .0.  )  /\  A. y  e.  X  ( N `  ( x 
 .-  y ) ) 
 <_  ( ( N `  x )  +  ( N `  y ) ) ) ) ) )
 
Theoremisnrg 18003 A normed ring is a ring with a norm that makes it into a normed group, and such that the norm is an absolute value on the ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  N  =  ( norm `  R )   &    |-  A  =  (AbsVal `  R )   =>    |-  ( R  e. NrmRing  <->  ( R  e. NrmGrp  /\  N  e.  A ) )
 
Theoremnrgabv 18004 The norm of a normed ring is an absolute value. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  N  =  ( norm `  R )   &    |-  A  =  (AbsVal `  R )   =>    |-  ( R  e. NrmRing  ->  N  e.  A )
 
Theoremnrgngp 18005 A normed ring is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( R  e. NrmRing  ->  R  e. NrmGrp )
 
Theoremnrgrng 18006 A normed ring is a ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( R  e. NrmRing  ->  R  e.  Ring )
 
Theoremnmmul 18007 The norm of a product in a normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  X  =  ( Base `  R )   &    |-  N  =  (
 norm `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e. NrmRing  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A  .x.  B ) )  =  ( ( N `
  A )  x.  ( N `  B ) ) )
 
Theoremnrgdsdi 18008 Distribute a distance calculation. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  X  =  ( Base `  R )   &    |-  N  =  (
 norm `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  D  =  ( dist `  R )   =>    |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( N `  A )  x.  ( B D C ) )  =  (
 ( A  .x.  B ) D ( A  .x.  C ) ) )
 
Theoremnrgdsdir 18009 Distribute a distance calculation. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  X  =  ( Base `  R )   &    |-  N  =  (
 norm `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  D  =  ( dist `  R )   =>    |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A D B )  x.  ( N `  C ) )  =  (
 ( A  .x.  C ) D ( B  .x.  C ) ) )
 
Theoremnm1 18010 The norm of one in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  N  =  ( norm `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e. NrmRing  /\  R  e. NzRing )  ->  ( N `  .1.  )  =  1 )
 
Theoremunitnmn0 18011 The norm of a unit is nonzero in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  N  =  ( norm `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( ( R  e. NrmRing  /\  R  e. NzRing  /\  A  e.  U )  ->  ( N `
  A )  =/=  0 )
 
Theoremnminvr 18012 The norm of an inverse in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  N  =  ( norm `  R )   &    |-  U  =  (Unit `  R )   &    |-  I  =  (
 invr `  R )   =>    |-  ( ( R  e. NrmRing  /\  R  e. NzRing  /\  A  e.  U )  ->  ( N `  ( I `  A ) )  =  ( 1  /  ( N `  A ) ) )
 
Theoremnmdvr 18013 The norm of a division in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  X  =  ( Base `  R )   &    |-  N  =  (
 norm `  R )   &    |-  U  =  (Unit `  R )   &    |-  ./  =  (/r `  R )   =>    |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  ->  ( N `  ( A  ./  B ) )  =  (
 ( N `  A )  /  ( N `  B ) ) )
 
Theoremnrgdomn 18014 A nonzero normed ring is a domain. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( R  e. NrmRing  ->  ( R  e. Domn  <->  R  e. NzRing ) )
 
Theoremnrgtgp 18015 A normed ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. NrmRing  ->  R  e.  TopGrp )
 
Theoremsubrgnrg 18016 A normed ring restricted to a subring is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  H  =  ( Gs  A )   =>    |-  ( ( G  e. NrmRing  /\  A  e.  (SubRing `  G ) )  ->  H  e. NrmRing )
 
Theoremtngnrg 18017 Given any absolute value over a ring, augmenting the ring with the absolute value produces a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  T  =  ( R toNrmGrp  F )   &    |-  A  =  (AbsVal `  R )   =>    |-  ( F  e.  A  ->  T  e. NrmRing )
 
Theoremisnlm 18018* A normed (left) module is a module which is also a normed group over a normed ring, such that the norm distributes over scalar multiplication. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 norm `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  A  =  (
 norm `  F )   =>    |-  ( W  e. NrmMod  <->  (
 ( W  e. NrmGrp  /\  W  e.  LMod  /\  F  e. NrmRing ) 
 /\  A. x  e.  K  A. y  e.  V  ( N `  ( x 
 .x.  y ) )  =  ( ( A `
  x )  x.  ( N `  y
 ) ) ) )
 
Theoremnmvs 18019 Defining property of a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 norm `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  A  =  (
 norm `  F )   =>    |-  ( ( W  e. NrmMod  /\  X  e.  K  /\  Y  e.  V ) 
 ->  ( N `  ( X  .x.  Y ) )  =  ( ( A `
  X )  x.  ( N `  Y ) ) )
 
Theoremnlmngp 18020 A normed module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( W  e. NrmMod  ->  W  e. NrmGrp )
 
Theoremnlmlmod 18021 A normed module is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( W  e. NrmMod  ->  W  e.  LMod )
 
Theoremnlmnrg 18022 The scalar component of a left module is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. NrmMod  ->  F  e. NrmRing )
 
Theoremnlmngp2 18023 The scalar component of a left module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. NrmMod  ->  F  e. NrmGrp )
 
Theoremnlmdsdi 18024 Distribute a distance calculation. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  D  =  ( dist `  W )   &    |-  A  =  ( norm `  F )   =>    |-  (
 ( W  e. NrmMod  /\  ( X  e.  K  /\  Y  e.  V  /\  Z  e.  V )
 )  ->  ( ( A `  X )  x.  ( Y D Z ) )  =  (
 ( X  .x.  Y ) D ( X  .x.  Z ) ) )
 
Theoremnlmdsdir 18025 Distribute a distance calculation. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  D  =  ( dist `  W )   &    |-  N  =  ( norm `  W )   &    |-  E  =  ( dist `  F )   =>    |-  (
 ( W  e. NrmMod  /\  ( X  e.  K  /\  Y  e.  K  /\  Z  e.  V )
 )  ->  ( ( X E Y )  x.  ( N `  Z ) )  =  (
 ( X  .x.  Z ) D ( Y  .x.  Z ) ) )
 
Theoremnlmmul0or 18026 If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 6-Dec-2007.) (Revised by Mario Carneiro, 4-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  O  =  ( 0g `  F )   =>    |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  ( ( A  .x.  B )  =  .0.  <->  ( A  =  O  \/  B  =  .0.  ) ) )
 
Theoremsranlm 18027 The subring algebra over a normed ring is a normed left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  A  =  ( ( subringAlg  `  W ) `  S )   =>    |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W ) )  ->  A  e. NrmMod )
 
Theoremnlmvscnlem2 18028 Lemma for nlmvscn 18030. Compare this proof with the similar elementary proof mulcn2 11946 for continuity of multiplication on  CC. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  V  =  ( Base `  W )   &    |-  K  =  (
 Base `  F )   &    |-  D  =  ( dist `  W )   &    |-  E  =  ( dist `  F )   &    |-  N  =  ( norm `  W )   &    |-  A  =  ( norm `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  T  =  ( ( R  /  2 )  /  ( ( A `  B )  +  1
 ) )   &    |-  U  =  ( ( R  /  2
 )  /  ( ( N `  X )  +  T ) )   &    |-  ( ph  ->  W  e. NrmMod )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  B  e.  K )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  C  e.  K )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  ( B E C )  <  U )   &    |-  ( ph  ->  ( X D Y )  <  T )   =>    |-  ( ph  ->  ( ( B  .x.  X ) D ( C  .x.  Y ) )  <  R )
 
Theoremnlmvscnlem1 18029* Lemma for nlmvscn 18030. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  V  =  ( Base `  W )   &    |-  K  =  (
 Base `  F )   &    |-  D  =  ( dist `  W )   &    |-  E  =  ( dist `  F )   &    |-  N  =  ( norm `  W )   &    |-  A  =  ( norm `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  T  =  ( ( R  /  2 )  /  ( ( A `  B )  +  1
 ) )   &    |-  U  =  ( ( R  /  2
 )  /  ( ( N `  X )  +  T ) )   &    |-  ( ph  ->  W  e. NrmMod )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  B  e.  K )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  E. r  e.  RR+  A. x  e.  K  A. y  e.  V  ( ( ( B E x )  <  r  /\  ( X D y )  < 
 r )  ->  (
 ( B  .x.  X ) D ( x  .x.  y ) )  <  R ) )
 
Theoremnlmvscn 18030 The scalar multiplication of a normed module is continuous. Lemma for nrgtrg 18032 and nlmtlm 18036. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .x.  =  ( .s f `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  K  =  ( TopOpen `  F )   =>    |-  ( W  e. NrmMod  ->  .x.  e.  ( ( K  tX  J )  Cn  J ) )
 
Theoremrlmnlm 18031 The ring module over a normed ring is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( R  e. NrmRing  ->  (ringLMod `  R )  e. NrmMod )
 
Theoremnrgtrg 18032 A normed ring is a topological ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( R  e. NrmRing  ->  R  e. 
 TopRing )
 
Theoremnrginvrcnlem 18033* Lemma for nrginvrcn 18034. Compare this proof with reccn2 11947, the elementary proof of continuity of division. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  X  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  I  =  (
 invr `  R )   &    |-  N  =  ( norm `  R )   &    |-  D  =  ( dist `  R )   &    |-  ( ph  ->  R  e. NrmRing )   &    |-  ( ph  ->  R  e. NzRing )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  T  =  ( if ( 1  <_  ( ( N `  A )  x.  B ) ,  1 ,  ( ( N `  A )  x.  B ) )  x.  (
 ( N `  A )  /  2 ) )   =>    |-  ( ph  ->  E. x  e.  RR+  A. y  e.  U  ( ( A D y )  <  x  ->  ( ( I `  A ) D ( I `  y ) )  <  B ) )
 
Theoremnrginvrcn 18034 The ring inverse function is continuous in a normed ring. (Note that this is true even in rings which are not division rings.) (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  X  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  I  =  (
 invr `  R )   &    |-  J  =  ( TopOpen `  R )   =>    |-  ( R  e. NrmRing  ->  I  e.  ( ( Jt  U )  Cn  ( Jt  U ) ) )
 
Theoremnrgtdrg 18035 A normed division ring is a topological division ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( ( R  e. NrmRing  /\  R  e.  DivRing )  ->  R  e. TopDRing )
 
Theoremnlmtlm 18036 A normed module is a topological module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( W  e. NrmMod  ->  W  e. TopMod )
 
Theoremisnvc 18037 A normed vector space is just a normed module which is algebraically a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( W  e. NrmVec  <->  ( W  e. NrmMod  /\  W  e.  LVec )
 )
 
Theoremnvcnlm 18038 A normed vector space is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( W  e. NrmVec  ->  W  e. NrmMod )
 
Theoremnvclvec 18039 A normed vector space is a left vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( W  e. NrmVec  ->  W  e.  LVec )
 
Theoremnvclmod 18040 A normed vector space is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( W  e. NrmVec  ->  W  e.  LMod )
 
Theoremisnvc2 18041 A normed vector space is just a normed module whose scalar ring is a division ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. NrmVec  <->  ( W  e. NrmMod  /\  F  e.  DivRing ) )
 
Theoremnvctvc 18042 A normed vector space is a topological vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( W  e. NrmVec  ->  W  e.  TopVec )
 
Theoremlssnlm 18043 A subspace of a normed module is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  X  =  ( Ws  U )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  X  e. NrmMod )
 
Theoremlssnvc 18044 A subspace of a normed vector space is a normed vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  X  =  ( Ws  U )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( ( W  e. NrmVec  /\  U  e.  S )  ->  X  e. NrmVec )
 
Theoremrlmnvc 18045 The ring module over a normed division ring is a normed vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( ( R  e. NrmRing  /\  R  e.  DivRing )  ->  (ringLMod `  R )  e. NrmVec )
 
11.3.7  Normed space homomorphisms (bounded linear operators)
 
Syntaxcnmo 18046 The operator norm function.
 class  normOp
 
Syntaxcnghm 18047 The class of normed group homomorphisms.
 class NGHom
 
Syntaxcnmhm 18048 The class of normed module homomorphisms.
 class NMHom
 
Definitiondf-nmo 18049* Define the norm of an operator between two normed groups (usually vector spaces). This definition produces an operator norm function for each pair of groups  <. s ,  t
>.. Equivalent to the definition of linear operator norm in [AkhiezerGlazman] p. 39. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  normOp  =  ( s  e. NrmGrp ,  t  e. NrmGrp  |->  ( f  e.  ( s  GrpHom  t )  |->  sup ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  ( Base `  s )
 ( ( norm `  t
 ) `  ( f `  x ) )  <_  ( r  x.  (
 ( norm `  s ) `  x ) ) } ,  RR* ,  `'  <  ) ) )
 
Definitiondf-nghm 18050* Define the set of normed group homomorphisms between two normed groups. A normed group homomorphism is a group homomorphism which additionally bounds the increase of norm by a fixed real operator. In vector spaces these are also known as bounded linear operators. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |- NGHom  =  ( s  e. NrmGrp ,  t  e. NrmGrp 
 |->  ( `' ( s
 normOp t ) " RR ) )
 
Definitiondf-nmhm 18051* Define a normed module homomorphism, also known as a bounded linear operator. This is a module homomorphism (a linear function) such that the operator norm is finite, or equivalently there is a constant  c such that (Contributed by Mario Carneiro, 18-Oct-2015.)
 |- NMHom  =  ( s  e. NrmMod ,  t  e. NrmMod 
 |->  ( ( s LMHom  t
 )  i^i  ( s NGHom  t ) ) )
 
Theoremnmoffn 18052 The function producing operator norm functions is a function on normed groups. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  normOp  Fn  (NrmGrp  X. NrmGrp )
 
Theoremreldmnghm 18053 Lemma for normed group homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |- 
 Rel  dom NGHom
 
Theoremreldmnmhm 18054 Lemma for module homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |- 
 Rel  dom NMHom
 
Theoremnmofval 18055* Value of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   =>    |-  (
 ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  N  =  ( f  e.  ( S  GrpHom  T )  |->  sup ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  V  ( M `  ( f `  x ) )  <_  ( r  x.  ( L `  x ) ) } ,  RR* ,  `'  <  ) ) )
 
Theoremnmoval 18056* Value of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   =>    |-  (
 ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) ) 
 ->  ( N `  F )  =  sup ( {
 r  e.  ( 0 [,)  +oo )  |  A. x  e.  V  ( M `  ( F `  x ) )  <_  ( r  x.  ( L `  x ) ) } ,  RR* ,  `'  <  ) )
 
Theoremnmogelb 18057* Property of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   =>    |-  (
 ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  A  e.  RR* )  ->  ( A  <_  ( N `  F ) 
 <-> 
 A. r  e.  (
 0 [,)  +oo ) (
 A. x  e.  V  ( M `  ( F `
  x ) ) 
 <_  ( r  x.  ( L `  x ) ) 
 ->  A  <_  r )
 ) )
 
Theoremnmolb 18058* Any upper bound on the values of a linear operator translates to an upper bound on the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   =>    |-  (
 ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  A  e.  RR  /\  0  <_  A )  ->  ( A. x  e.  V  ( M `  ( F `  x ) )  <_  ( A  x.  ( L `  x ) )  ->  ( N `
  F )  <_  A ) )
 
Theoremnmolb2d 18059* Any upper bound on the values of a linear operator at nonzero vectors translates to an upper bound on the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   &    |-  .0.  =  ( 0g `  S )   &    |-  ( ph  ->  S  e. NrmGrp )   &    |-  ( ph  ->  T  e. NrmGrp )   &    |-  ( ph  ->  F  e.  ( S  GrpHom  T ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ( ph  /\  ( x  e.  V  /\  x  =/=  .0.  ) ) 
 ->  ( M `  ( F `  x ) ) 
 <_  ( A  x.  ( L `  x ) ) )   =>    |-  ( ph  ->  ( N `  F )  <_  A )
 
Theoremnmof 18060 The operator norm is a function into the extended reals. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   =>    |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  N : ( S  GrpHom  T ) --> RR* )
 
Theoremnmocl 18061 The operator norm of an operator is an extended real. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   =>    |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( N `  F )  e.  RR* )
 
Theoremnmoge0 18062 The operator norm of an operator is nonnegative. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   =>    |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  0  <_  ( N `  F ) )
 
Theoremnghmfval 18063 A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   =>    |-  ( S NGHom  T )  =  ( `' N " RR )
 
Theoremisnghm 18064 A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   =>    |-  ( F  e.  ( S NGHom  T )  <->  ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( F  e.  ( S  GrpHom  T )  /\  ( N `  F )  e.  RR ) ) )
 
Theoremisnghm2 18065 A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   =>    |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( F  e.  ( S NGHom  T )  <-> 
 ( N `  F )  e.  RR )
 )
 
Theoremisnghm3 18066 A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   =>    |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( F  e.  ( S NGHom  T )  <-> 
 ( N `  F )  <  +oo ) )
 
Theorembddnghm 18067 A bounded group homomorphism is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   =>    |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( A  e.  RR  /\  ( N `  F )  <_  A ) )  ->  F  e.  ( S NGHom  T ) )
 
Theoremnghmcl 18068 A normed group homomorphism has a real operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   =>    |-  ( F  e.  ( S NGHom  T )  ->  ( N `  F )  e. 
 RR )
 
Theoremnmoi 18069 The operator norm achieves the minimum of the set of upper bounds, if the operator is bounded. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   =>    |-  (
 ( F  e.  ( S NGHom  T )  /\  X  e.  V )  ->  ( M `  ( F `  X ) )  <_  ( ( N `  F )  x.  ( L `  X ) ) )
 
Theoremnmoix 18070 The operator norm is a bound on the size of an operator, even when it is infinite (using extended real multiplication). (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   =>    |-  (
 ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  X  e.  V )  ->  ( M `
  ( F `  X ) )  <_  ( ( N `  F ) x e
 ( L `  X ) ) )
 
Theoremnmoi2 18071 The operator norm is a bound on the growth of a vector under the action of the operator. (Contributed by Mario Carneiro, 19-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   &    |-  .0.  =  ( 0g `  S )   =>    |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/=  .0.  ) )  ->  ( ( M `  ( F `  X ) )  /  ( L `
  X ) ) 
 <_  ( N `  F ) )
 
Theoremnmoleub 18072* The operator norm, defined as an infimum of upper bounds, can also be defined as a supremum of norms of  F ( x ) away from zero. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   &    |-  .0.  =  ( 0g `  S )   &    |-  ( ph  ->  S  e. NrmGrp )   &    |-  ( ph  ->  T  e. NrmGrp )   &    |-  ( ph  ->  F  e.  ( S  GrpHom  T ) )   &    |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  (
 ( N `  F )  <_  A  <->  A. x  e.  V  ( x  =/=  .0.  ->  ( ( M `  ( F `  x ) ) 
 /  ( L `  x ) )  <_  A ) ) )
 
Theoremnghmrcl1 18073 Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  ( F  e.  ( S NGHom  T )  ->  S  e. NrmGrp )
 
Theoremnghmrcl2 18074 Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  ( F  e.  ( S NGHom  T )  ->  T  e. NrmGrp )
 
Theoremnghmghm 18075 A normed group homomorphism is a group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  ( F  e.  ( S NGHom  T )  ->  F  e.  ( S  GrpHom  T ) )
 
Theoremnmo0 18076 The operator norm of the zero operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  T )   =>    |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( N `  ( V  X.  {  .0.  } ) )  =  0 )
 
Theoremnmoeq0 18077 The operator norm is zero only for the zero operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  T )   =>    |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( ( N `  F )  =  0  <->  F  =  ( V  X.  {  .0.  }
 ) ) )
 
Theoremnmoco 18078 An upper bound on the operator norm of a composition. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  N  =  ( S
 normOp U )   &    |-  L  =  ( T normOp U )   &    |-  M  =  ( S normOp T )   =>    |-  ( ( F  e.  ( T NGHom  U )  /\  G  e.  ( S NGHom  T ) )  ->  ( N `  ( F  o.  G ) )  <_  ( ( L `  F )  x.  ( M `  G ) ) )
 
Theoremnghmco 18079 The composition of normed group homomorphisms is a normed group homomorphism. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  ( ( F  e.  ( T NGHom  U )  /\  G  e.  ( S NGHom  T ) )  ->  ( F  o.  G )  e.  ( S NGHom  U ) )
 
Theoremnmotri 18080 Triangle inequality for the operator norm. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  .+  =  ( +g  `  T )   =>    |-  ( ( T  e.  Abel  /\  F  e.  ( S NGHom  T )  /\  G  e.  ( S NGHom  T ) )  ->  ( N `  ( F  o F  .+  G ) ) 
 <_  ( ( N `  F )  +  ( N `  G ) ) )
 
Theoremnghmplusg 18081 The sum of two bounded linear operators is bounded linear. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |- 
 .+  =  ( +g  `  T )   =>    |-  ( ( T  e.  Abel  /\  F  e.  ( S NGHom  T )  /\  G  e.  ( S NGHom  T ) ) 
 ->  ( F  o F  .+  G )  e.  ( S NGHom  T ) )
 
Theorem0nghm 18082 The zero operator is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  V  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  T )   =>    |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( V  X.  {  .0.  }
 )  e.  ( S NGHom  T ) )
 
Theoremnmoid 18083 The operator norm of the identity function on a nontrivial group. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  N  =  ( S
 normOp S )   &    |-  V  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  S )   =>    |-  ( ( S  e. NrmGrp  /\ 
 {  .0.  }  C.  V )  ->  ( N `  (  _I  |`  V )
 )  =  1 )
 
Theoremidnghm 18084 The identity operator is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  V  =  ( Base `  S )   =>    |-  ( S  e. NrmGrp  ->  (  _I  |`  V )  e.  ( S NGHom  S ) )
 
Theoremnmods 18085 Upper bound for the distance between the values of a bounded linear operator. (Contributed by Mario Carneiro, 22-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  C  =  ( dist `  S )   &    |-  D  =  ( dist `  T )   =>    |-  (
 ( F  e.  ( S NGHom  T )  /\  A  e.  V  /\  B  e.  V )  ->  ( ( F `  A ) D ( F `  B ) )  <_  ( ( N `  F )  x.  ( A C B ) ) )
 
Theoremnghmcn 18086 A normed group homomorphism is a continuous function. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  J  =  ( TopOpen `  S )   &    |-  K  =  (
 TopOpen `  T )   =>    |-  ( F  e.  ( S NGHom  T )  ->  F  e.  ( J  Cn  K ) )
 
Theoremisnmhm 18087 A normed module homomorphism is a left module homomorphism which is also a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  ( F  e.  ( S NMHom  T )  <->  ( ( S  e. NrmMod  /\  T  e. NrmMod )  /\  ( F  e.  ( S LMHom  T )  /\  F  e.  ( S NGHom  T ) ) ) )
 
Theoremnmhmrcl1 18088 Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  ( F  e.  ( S NMHom  T )  ->  S  e. NrmMod )
 
Theoremnmhmrcl2 18089 Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  ( F  e.  ( S NMHom  T )  ->  T  e. NrmMod )
 
Theoremnmhmlmhm 18090 A normed module homomorphism is a left module homomorphism (a linear operator). (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  ( F  e.  ( S NMHom  T )  ->  F  e.  ( S LMHom  T ) )
 
Theoremnmhmnghm 18091 A normed module homomorphism is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  ( F  e.  ( S NMHom  T )  ->  F  e.  ( S NGHom  T ) )
 
Theoremnmhmghm 18092 A normed module homomorphism is a group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  ( F  e.  ( S NMHom  T )  ->  F  e.  ( S  GrpHom  T ) )
 
Theoremisnmhm2 18093 A normed module homomorphism is a left module homomorphism with bounded norm (a bounded linear operator). (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   =>    |-  ( ( S  e. NrmMod  /\  T  e. NrmMod  /\  F  e.  ( S LMHom  T ) ) 
 ->  ( F  e.  ( S NMHom  T )  <->  ( N `  F )  e.  RR ) )
 
Theoremnmhmcl 18094 A normed module homomorphism has a real operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   =>    |-  ( F  e.  ( S NMHom  T )  ->  ( N `  F )  e. 
 RR )
 
Theoremidnmhm 18095 The identity operator is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  V  =  ( Base `  S )   =>    |-  ( S  e. NrmMod  ->  (  _I  |`  V )  e.  ( S NMHom  S ) )
 
Theorem0nmhm 18096 The zero operator is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  V  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  T )   &    |-  F  =  (Scalar `  S )   &    |-  G  =  (Scalar `  T )   =>    |-  ( ( S  e. NrmMod  /\  T  e. NrmMod  /\  F  =  G )  ->  ( V  X.  {  .0.  }
 )  e.  ( S NMHom  T ) )
 
Theoremnmhmco 18097 The composition of bounded linear operators is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  ( ( F  e.  ( T NMHom  U )  /\  G  e.  ( S NMHom  T ) )  ->  ( F  o.  G )  e.  ( S NMHom  U ) )
 
Theoremnmhmplusg 18098 The sum of two bounded linear operators is bounded linear. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |- 
 .+  =  ( +g  `  T )   =>    |-  ( ( F  e.  ( S NMHom  T )  /\  G  e.  ( S NMHom  T ) )  ->  ( F  o F  .+  G )  e.  ( S NMHom  T ) )
 
11.3.8  Topology on the Reals
 
Theoremqtopbaslem 18099 The set of open intervals with endpoints in a subset forms a basis for a topology. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  S  C_  RR*   =>    |-  ( (,) " ( S  X.  S ) )  e.  TopBases
 
Theoremqtopbas 18100 The set of open intervals with rational endpoints forms a basis for a topology. (Contributed by NM, 8-Mar-2007.)
 |-  ( (,) " ( QQ  X.  QQ ) )  e.  TopBases
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