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Definition df-nmoo 24167
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  |-  normOpOLD  =  ( 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 24163 . 2  class  normOpOLD
2 vu . . 3  setvar  u
3 vw . . 3  setvar  w
4 cnv 23984 . . 3  class  NrmCVec
5 vt . . . 4  setvar  t
63cv 1368 . . . . . 6  class  w
7 cba 23986 . . . . . 6  class  BaseSet
86, 7cfv 5439 . . . . 5  class  ( BaseSet `  w )
92cv 1368 . . . . . 6  class  u
109, 7cfv 5439 . . . . 5  class  ( BaseSet `  u )
11 cmap 7235 . . . . 5  class  ^m
128, 10, 11co 6112 . . . 4  class  ( (
BaseSet `  w )  ^m  ( BaseSet `  u )
)
13 vz . . . . . . . . . . 11  setvar  z
1413cv 1368 . . . . . . . . . 10  class  z
15 cnmcv 23990 . . . . . . . . . . 11  class  normCV
169, 15cfv 5439 . . . . . . . . . 10  class  ( normCV `  u )
1714, 16cfv 5439 . . . . . . . . 9  class  ( (
normCV
`  u ) `  z )
18 c1 9304 . . . . . . . . 9  class  1
19 cle 9440 . . . . . . . . 9  class  <_
2017, 18, 19wbr 4313 . . . . . . . 8  wff  ( (
normCV
`  u ) `  z )  <_  1
21 vx . . . . . . . . . 10  setvar  x
2221cv 1368 . . . . . . . . 9  class  x
235cv 1368 . . . . . . . . . . 11  class  t
2414, 23cfv 5439 . . . . . . . . . 10  class  ( t `
 z )
256, 15cfv 5439 . . . . . . . . . 10  class  ( normCV `  w )
2624, 25cfv 5439 . . . . . . . . 9  class  ( (
normCV
`  w ) `  ( t `  z
) )
2722, 26wceq 1369 . . . . . . . 8  wff  x  =  ( ( normCV `  w
) `  ( t `  z ) )
2820, 27wa 369 . . . . . . 7  wff  ( ( ( normCV `  u ) `  z )  <_  1  /\  x  =  (
( normCV `  w ) `  ( t `  z
) ) )
2928, 13, 10wrex 2737 . . . . . 6  wff  E. z  e.  ( BaseSet `  u )
( ( ( normCV `  u ) `  z
)  <_  1  /\  x  =  ( ( normCV `  w ) `  (
t `  z )
) )
3029, 21cab 2429 . . . . 5  class  { x  |  E. z  e.  (
BaseSet `  u ) ( ( ( normCV `  u
) `  z )  <_  1  /\  x  =  ( ( normCV `  w
) `  ( t `  z ) ) ) }
31 cxr 9438 . . . . 5  class  RR*
32 clt 9439 . . . . 5  class  <
3330, 31, 32csup 7711 . . . 4  class  sup ( { x  |  E. z  e.  ( BaseSet `  u ) ( ( ( normCV `  u ) `  z )  <_  1  /\  x  =  (
( normCV `  w ) `  ( t `  z
) ) ) } ,  RR* ,  <  )
345, 12, 33cmpt 4371 . . 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 6114 . 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 1369 1  wff  normOpOLD  =  ( 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 setvar class
This definition is referenced by:  nmoofval  24184
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