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Definition df-nmoo 21339
Description: 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.)
Assertion
Ref Expression
df-nmoo  |-  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* ,  <  ) ) )
Distinct variable group:    u, t, w, x, z

Detailed syntax breakdown of Definition df-nmoo
StepHypRef Expression
1 cnmoo 21335 . 2  class  normOp OLD
2 vu . . 3  set  u
3 vw . . 3  set  w
4 cnv 21156 . . 3  class  NrmCVec
5 vt . . . 4  set  t
63cv 1631 . . . . . 6  class  w
7 cba 21158 . . . . . 6  class  BaseSet
86, 7cfv 5271 . . . . 5  class  ( BaseSet `  w )
92cv 1631 . . . . . 6  class  u
109, 7cfv 5271 . . . . 5  class  ( BaseSet `  u )
11 cmap 6788 . . . . 5  class  ^m
128, 10, 11co 5874 . . . 4  class  ( (
BaseSet `  w )  ^m  ( BaseSet `  u )
)
13 vz . . . . . . . . . . 11  set  z
1413cv 1631 . . . . . . . . . 10  class  z
15 cnmcv 21162 . . . . . . . . . . 11  class  normCV
169, 15cfv 5271 . . . . . . . . . 10  class  ( normCV `  u )
1714, 16cfv 5271 . . . . . . . . 9  class  ( (
normCV
`  u ) `  z )
18 c1 8754 . . . . . . . . 9  class  1
19 cle 8884 . . . . . . . . 9  class  <_
2017, 18, 19wbr 4039 . . . . . . . 8  wff  ( (
normCV
`  u ) `  z )  <_  1
21 vx . . . . . . . . . 10  set  x
2221cv 1631 . . . . . . . . 9  class  x
235cv 1631 . . . . . . . . . . 11  class  t
2414, 23cfv 5271 . . . . . . . . . 10  class  ( t `
 z )
256, 15cfv 5271 . . . . . . . . . 10  class  ( normCV `  w )
2624, 25cfv 5271 . . . . . . . . 9  class  ( (
normCV
`  w ) `  ( t `  z
) )
2722, 26wceq 1632 . . . . . . . 8  wff  x  =  ( ( normCV `  w
) `  ( t `  z ) )
2820, 27wa 358 . . . . . . 7  wff  ( ( ( normCV `  u ) `  z )  <_  1  /\  x  =  (
( normCV `  w ) `  ( t `  z
) ) )
2928, 13, 10wrex 2557 . . . . . 6  wff  E. z  e.  ( BaseSet `  u )
( ( ( normCV `  u ) `  z
)  <_  1  /\  x  =  ( ( normCV `  w ) `  (
t `  z )
) )
3029, 21cab 2282 . . . . 5  class  { x  |  E. z  e.  (
BaseSet `  u ) ( ( ( normCV `  u
) `  z )  <_  1  /\  x  =  ( ( normCV `  w
) `  ( t `  z ) ) ) }
31 cxr 8882 . . . . 5  class  RR*
32 clt 8883 . . . . 5  class  <
3330, 31, 32csup 7209 . . . 4  class  sup ( { x  |  E. z  e.  ( BaseSet `  u ) ( ( ( normCV `  u ) `  z )  <_  1  /\  x  =  (
( normCV `  w ) `  ( t `  z
) ) ) } ,  RR* ,  <  )
345, 12, 33cmpt 4093 . . 3  class  ( t  e.  ( ( BaseSet `  w )  ^m  ( BaseSet
`  u ) ) 
|->  sup ( { x  |  E. z  e.  (
BaseSet `  u ) ( ( ( normCV `  u
) `  z )  <_  1  /\  x  =  ( ( normCV `  w
) `  ( t `  z ) ) ) } ,  RR* ,  <  ) )
352, 3, 4, 4, 34cmpt2 5876 . 2  class  ( 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* ,  <  ) ) )
361, 35wceq 1632 1  wff  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* ,  <  ) ) )
Colors of variables: wff set class
This definition is referenced by:  nmoofval  21356
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