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Theorem List for Metamath Proof Explorer - 18201-18300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtngnrg 18201 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 18202* 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 18203 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 18204 A normed module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( W  e. NrmMod  ->  W  e. NrmGrp )
 
Theoremnlmlmod 18205 A normed module is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( W  e. NrmMod  ->  W  e.  LMod )
 
Theoremnlmnrg 18206 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 18207 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 18208 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 18209 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 18210 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 18211 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 18212 Lemma for nlmvscn 18214. Compare this proof with the similar elementary proof mulcn2 12085 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 18213* Lemma for nlmvscn 18214. (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 18214 The scalar multiplication of a normed module is continuous. Lemma for nrgtrg 18216 and nlmtlm 18220. (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 18215 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 18216 A normed ring is a topological ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( R  e. NrmRing  ->  R  e. 
 TopRing )
 
Theoremnrginvrcnlem 18217* Lemma for nrginvrcn 18218. Compare this proof with reccn2 12086, 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 18218 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 18219 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 18220 A normed module is a topological module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( W  e. NrmMod  ->  W  e. TopMod )
 
Theoremisnvc 18221 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 18222 A normed vector space is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( W  e. NrmVec  ->  W  e. NrmMod )
 
Theoremnvclvec 18223 A normed vector space is a left vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( W  e. NrmVec  ->  W  e.  LVec )
 
Theoremnvclmod 18224 A normed vector space is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( W  e. NrmVec  ->  W  e.  LMod )
 
Theoremisnvc2 18225 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 18226 A normed vector space is a topological vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( W  e. NrmVec  ->  W  e.  TopVec )
 
Theoremlssnlm 18227 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 18228 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 18229 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 18230 The operator norm function.
 class  normOp
 
Syntaxcnghm 18231 The class of normed group homomorphisms.
 class NGHom
 
Syntaxcnmhm 18232 The class of normed module homomorphisms.
 class NMHom
 
Definitiondf-nmo 18233* 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 18234* 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 18235* 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 18236 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 18237 Lemma for normed group homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |- 
 Rel  dom NGHom
 
Theoremreldmnmhm 18238 Lemma for module homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |- 
 Rel  dom NMHom
 
Theoremnmofval 18239* 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 18240* 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 18241* 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 18242* 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 18243* 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 18244 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 18245 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 18246 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 18247 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 18248 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 18249 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 18250 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 18251 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 18252 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 18253 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 18254 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 18255 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 18256* 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 18257 Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  ( F  e.  ( S NGHom  T )  ->  S  e. NrmGrp )
 
Theoremnghmrcl2 18258 Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  ( F  e.  ( S NGHom  T )  ->  T  e. NrmGrp )
 
Theoremnghmghm 18259 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 18260 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 18261 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 18262 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 18263 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 18264 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 18265 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 18266 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 18267 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 18268 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 18269 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 18270 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 18271 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 18272 Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  ( F  e.  ( S NMHom  T )  ->  S  e. NrmMod )
 
Theoremnmhmrcl2 18273 Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  ( F  e.  ( S NMHom  T )  ->  T  e. NrmMod )
 
Theoremnmhmlmhm 18274 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 18275 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 18276 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 18277 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 18278 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 18279 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 18280 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 18281 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 18282 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 18283 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 18284 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
 
Theoremretopbas 18285 A basis for the standard topology on the reals. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 17-Jun-2014.)
 |- 
 ran  (,)  e.  TopBases
 
Theoremretop 18286 The standard topology on the reals. (Contributed by FL, 4-Jun-2007.)
 |-  ( topGen `  ran  (,) )  e.  Top
 
Theoremuniretop 18287 The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007.)
 |- 
 RR  =  U. ( topGen `
  ran  (,) )
 
Theoremretopon 18288 The standard topology on the reals is a topology on the reals. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( topGen `  ran  (,) )  e.  (TopOn `  RR )
 
TheoremretpsOLD 18289 The standard topological space on the reals. (Contributed by NM, 10-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 <. RR ,  ( topGen `  ran  (,) ) >.  e.  TopSp OLD
 
Theoremretps 18290 The standard topological space on the reals. (Contributed by NM, 19-Oct-2012.)
 |-  K  =  { <. (
 Base `  ndx ) ,  RR >. ,  <. (TopSet `  ndx ) ,  ( topGen `  ran  (,) ) >. }   =>    |-  K  e.  TopSp
 
Theoremiooretop 18291 Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.)
 |-  ( A (,) B )  e.  ( topGen `  ran  (,) )
 
Theoremicccld 18292 Closed intervals are closed sets of the standard topology on  RR. (Contributed by FL, 14-Sep-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B )  e.  ( Clsd `  ( topGen `  ran  (,) )
 ) )
 
Theoremicopnfcld 18293 Right-unbounded closed intervals are closed sets of the standard topology on  RR. (Contributed by Mario Carneiro, 17-Feb-2015.)
 |-  ( A  e.  RR  ->  ( A [,)  +oo )  e.  ( Clsd `  ( topGen `  ran  (,) )
 ) )
 
Theoremiocmnfcld 18294 Left-unbounded closed intervals are closed sets of the standard topology on  RR. (Contributed by Mario Carneiro, 17-Feb-2015.)
 |-  ( A  e.  RR  ->  (  -oo (,] A )  e.  ( Clsd `  ( topGen `  ran  (,) )
 ) )
 
Theoremqdensere 18295  QQ is dense in the standard topology on  RR. (Contributed by NM, 1-Mar-2007.)
 |-  ( ( cls `  ( topGen `
  ran  (,) ) ) `
  QQ )  =  RR
 
Theoremcnmetdval 18296 Value of the distance function of the metric space of complex numbers. (Contributed by NM, 9-Dec-2006.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  D  =  ( abs 
 o.  -  )   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A D B )  =  ( abs `  ( A  -  B ) ) )
 
Theoremcnmet 18297 The absolute value metric determines a metric space on the complex numbers. This theorem provides a link between complex numbers and metrics spaces, making metric space theorems available for use with complex numbers. (Contributed by FL, 9-Oct-2006.)
 |-  ( abs  o.  -  )  e.  ( Met `  CC )
 
Theoremcnxmet 18298 The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( abs  o.  -  )  e.  ( * Met `  CC )
 
Theoremcnbl0 18299 Two ways to write the open ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)
 |-  D  =  ( abs 
 o.  -  )   =>    |-  ( R  e.  RR* 
 ->  ( `' abs " (
 0 [,) R ) )  =  ( 0 (
 ball `  D ) R ) )
 
Theoremcnblcld 18300* Two ways to write the closed ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)
 |-  D  =  ( abs 
 o.  -  )   =>    |-  ( R  e.  RR* 
 ->  ( `' abs " (
 0 [,] R ) )  =  { x  e. 
 CC  |  ( 0 D x )  <_  R } )
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