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Definition df-nmop 25243
Description: Define the norm of a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
Assertion
Ref Expression
df-nmop  |-  normop  =  ( t  e.  ( ~H 
^m  ~H )  |->  sup ( { x  |  E. z  e.  ~H  (
( normh `  z )  <_  1  /\  x  =  ( normh `  ( t `  z ) ) ) } ,  RR* ,  <  ) )
Distinct variable group:    x, t, z

Detailed syntax breakdown of Definition df-nmop
StepHypRef Expression
1 cnop 24347 . 2  class  normop
2 vt . . 3  setvar  t
3 chil 24321 . . . 4  class  ~H
4 cmap 7214 . . . 4  class  ^m
53, 3, 4co 6091 . . 3  class  ( ~H 
^m  ~H )
6 vz . . . . . . . . . 10  setvar  z
76cv 1368 . . . . . . . . 9  class  z
8 cno 24325 . . . . . . . . 9  class  normh
97, 8cfv 5418 . . . . . . . 8  class  ( normh `  z )
10 c1 9283 . . . . . . . 8  class  1
11 cle 9419 . . . . . . . 8  class  <_
129, 10, 11wbr 4292 . . . . . . 7  wff  ( normh `  z )  <_  1
13 vx . . . . . . . . 9  setvar  x
1413cv 1368 . . . . . . . 8  class  x
152cv 1368 . . . . . . . . . 10  class  t
167, 15cfv 5418 . . . . . . . . 9  class  ( t `
 z )
1716, 8cfv 5418 . . . . . . . 8  class  ( normh `  ( t `  z
) )
1814, 17wceq 1369 . . . . . . 7  wff  x  =  ( normh `  ( t `  z ) )
1912, 18wa 369 . . . . . 6  wff  ( (
normh `  z )  <_ 
1  /\  x  =  ( normh `  ( t `  z ) ) )
2019, 6, 3wrex 2716 . . . . 5  wff  E. z  e.  ~H  ( ( normh `  z )  <_  1  /\  x  =  ( normh `  ( t `  z ) ) )
2120, 13cab 2429 . . . 4  class  { x  |  E. z  e.  ~H  ( ( normh `  z
)  <_  1  /\  x  =  ( normh `  ( t `  z
) ) ) }
22 cxr 9417 . . . 4  class  RR*
23 clt 9418 . . . 4  class  <
2421, 22, 23csup 7690 . . 3  class  sup ( { x  |  E. z  e.  ~H  (
( normh `  z )  <_  1  /\  x  =  ( normh `  ( t `  z ) ) ) } ,  RR* ,  <  )
252, 5, 24cmpt 4350 . 2  class  ( t  e.  ( ~H  ^m  ~H )  |->  sup ( { x  |  E. z  e.  ~H  (
( normh `  z )  <_  1  /\  x  =  ( normh `  ( t `  z ) ) ) } ,  RR* ,  <  ) )
261, 25wceq 1369 1  wff  normop  =  ( t  e.  ( ~H 
^m  ~H )  |->  sup ( { x  |  E. z  e.  ~H  (
( normh `  z )  <_  1  /\  x  =  ( normh `  ( t `  z ) ) ) } ,  RR* ,  <  ) )
Colors of variables: wff setvar class
This definition is referenced by:  nmopval  25260  hhnmoi  25305
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