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Definition df-nmoo 21316
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 21312 . 2  class  normOp OLD
2 vu . . 3  set  u
3 vw . . 3  set  w
4 cnv 21133 . . 3  class  NrmCVec
5 vt . . . 4  set  t
63cv 1623 . . . . . 6  class  w
7 cba 21135 . . . . . 6  class  BaseSet
86, 7cfv 5222 . . . . 5  class  ( BaseSet `  w )
92cv 1623 . . . . . 6  class  u
109, 7cfv 5222 . . . . 5  class  ( BaseSet `  u )
11 cmap 6768 . . . . 5  class  ^m
128, 10, 11co 5820 . . . 4  class  ( (
BaseSet `  w )  ^m  ( BaseSet `  u )
)
13 vz . . . . . . . . . . 11  set  z
1413cv 1623 . . . . . . . . . 10  class  z
15 cnmcv 21139 . . . . . . . . . . 11  class  normCV
169, 15cfv 5222 . . . . . . . . . 10  class  ( normCV `  u )
1714, 16cfv 5222 . . . . . . . . 9  class  ( (
normCV
`  u ) `  z )
18 c1 8734 . . . . . . . . 9  class  1
19 cle 8864 . . . . . . . . 9  class  <_
2017, 18, 19wbr 4025 . . . . . . . 8  wff  ( (
normCV
`  u ) `  z )  <_  1
21 vx . . . . . . . . . 10  set  x
2221cv 1623 . . . . . . . . 9  class  x
235cv 1623 . . . . . . . . . . 11  class  t
2414, 23cfv 5222 . . . . . . . . . 10  class  ( t `
 z )
256, 15cfv 5222 . . . . . . . . . 10  class  ( normCV `  w )
2624, 25cfv 5222 . . . . . . . . 9  class  ( (
normCV
`  w ) `  ( t `  z
) )
2722, 26wceq 1624 . . . . . . . 8  wff  x  =  ( ( normCV `  w
) `  ( t `  z ) )
2820, 27wa 360 . . . . . . 7  wff  ( ( ( normCV `  u ) `  z )  <_  1  /\  x  =  (
( normCV `  w ) `  ( t `  z
) ) )
2928, 13, 10wrex 2546 . . . . . 6  wff  E. z  e.  ( BaseSet `  u )
( ( ( normCV `  u ) `  z
)  <_  1  /\  x  =  ( ( normCV `  w ) `  (
t `  z )
) )
3029, 21cab 2271 . . . . 5  class  { x  |  E. z  e.  (
BaseSet `  u ) ( ( ( normCV `  u
) `  z )  <_  1  /\  x  =  ( ( normCV `  w
) `  ( t `  z ) ) ) }
31 cxr 8862 . . . . 5  class  RR*
32 clt 8863 . . . . 5  class  <
3330, 31, 32csup 7189 . . . 4  class  sup ( { x  |  E. z  e.  ( BaseSet `  u ) ( ( ( normCV `  u ) `  z )  <_  1  /\  x  =  (
( normCV `  w ) `  ( t `  z
) ) ) } ,  RR* ,  <  )
345, 12, 33cmpt 4079 . . 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 5822 . 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 1624 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  21333
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