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Definition df-nmoo 21248
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 21244 . 2  class  normOp OLD
2 vu . . 3  set  u
3 vw . . 3  set  w
4 cnv 21065 . . 3  class  NrmCVec
5 vt . . . 4  set  t
63cv 1618 . . . . . 6  class  w
7 cba 21067 . . . . . 6  class  BaseSet
86, 7cfv 4638 . . . . 5  class  ( BaseSet `  w )
92cv 1618 . . . . . 6  class  u
109, 7cfv 4638 . . . . 5  class  ( BaseSet `  u )
11 cmap 6705 . . . . 5  class  ^m
128, 10, 11co 5757 . . . 4  class  ( (
BaseSet `  w )  ^m  ( BaseSet `  u )
)
13 vz . . . . . . . . . . 11  set  z
1413cv 1618 . . . . . . . . . 10  class  z
15 cnmcv 21071 . . . . . . . . . . 11  class  normCV
169, 15cfv 4638 . . . . . . . . . 10  class  ( normCV `  u )
1714, 16cfv 4638 . . . . . . . . 9  class  ( (
normCV
`  u ) `  z )
18 c1 8671 . . . . . . . . 9  class  1
19 cle 8801 . . . . . . . . 9  class  <_
2017, 18, 19wbr 3963 . . . . . . . 8  wff  ( (
normCV
`  u ) `  z )  <_  1
21 vx . . . . . . . . . 10  set  x
2221cv 1618 . . . . . . . . 9  class  x
235cv 1618 . . . . . . . . . . 11  class  t
2414, 23cfv 4638 . . . . . . . . . 10  class  ( t `
 z )
256, 15cfv 4638 . . . . . . . . . 10  class  ( normCV `  w )
2624, 25cfv 4638 . . . . . . . . 9  class  ( (
normCV
`  w ) `  ( t `  z
) )
2722, 26wceq 1619 . . . . . . . 8  wff  x  =  ( ( normCV `  w
) `  ( t `  z ) )
2820, 27wa 360 . . . . . . 7  wff  ( ( ( normCV `  u ) `  z )  <_  1  /\  x  =  (
( normCV `  w ) `  ( t `  z
) ) )
2928, 13, 10wrex 2517 . . . . . 6  wff  E. z  e.  ( BaseSet `  u )
( ( ( normCV `  u ) `  z
)  <_  1  /\  x  =  ( ( normCV `  w ) `  (
t `  z )
) )
3029, 21cab 2242 . . . . 5  class  { x  |  E. z  e.  (
BaseSet `  u ) ( ( ( normCV `  u
) `  z )  <_  1  /\  x  =  ( ( normCV `  w
) `  ( t `  z ) ) ) }
31 cxr 8799 . . . . 5  class  RR*
32 clt 8800 . . . . 5  class  <
3330, 31, 32csup 7126 . . . 4  class  sup ( { x  |  E. z  e.  ( BaseSet `  u ) ( ( ( normCV `  u ) `  z )  <_  1  /\  x  =  (
( normCV `  w ) `  ( t `  z
) ) ) } ,  RR* ,  <  )
345, 12, 33cmpt 4017 . . 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 5759 . 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 1619 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  21265
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