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Definition df-nmfn 23301
Description: Define the norm of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
df-nmfn  |-  normfn  =  ( t  e.  ( CC 
^m  ~H )  |->  sup ( { x  |  E. z  e.  ~H  (
( normh `  z )  <_  1  /\  x  =  ( abs `  (
t `  z )
) ) } ,  RR* ,  <  ) )
Distinct variable group:    x, t, z

Detailed syntax breakdown of Definition df-nmfn
StepHypRef Expression
1 cnmf 22407 . 2  class  normfn
2 vt . . 3  set  t
3 cc 8944 . . . 4  class  CC
4 chil 22375 . . . 4  class  ~H
5 cmap 6977 . . . 4  class  ^m
63, 4, 5co 6040 . . 3  class  ( CC 
^m  ~H )
7 vz . . . . . . . . . 10  set  z
87cv 1648 . . . . . . . . 9  class  z
9 cno 22379 . . . . . . . . 9  class  normh
108, 9cfv 5413 . . . . . . . 8  class  ( normh `  z )
11 c1 8947 . . . . . . . 8  class  1
12 cle 9077 . . . . . . . 8  class  <_
1310, 11, 12wbr 4172 . . . . . . 7  wff  ( normh `  z )  <_  1
14 vx . . . . . . . . 9  set  x
1514cv 1648 . . . . . . . 8  class  x
162cv 1648 . . . . . . . . . 10  class  t
178, 16cfv 5413 . . . . . . . . 9  class  ( t `
 z )
18 cabs 11994 . . . . . . . . 9  class  abs
1917, 18cfv 5413 . . . . . . . 8  class  ( abs `  ( t `  z
) )
2015, 19wceq 1649 . . . . . . 7  wff  x  =  ( abs `  (
t `  z )
)
2113, 20wa 359 . . . . . 6  wff  ( (
normh `  z )  <_ 
1  /\  x  =  ( abs `  ( t `
 z ) ) )
2221, 7, 4wrex 2667 . . . . 5  wff  E. z  e.  ~H  ( ( normh `  z )  <_  1  /\  x  =  ( abs `  ( t `  z ) ) )
2322, 14cab 2390 . . . 4  class  { x  |  E. z  e.  ~H  ( ( normh `  z
)  <_  1  /\  x  =  ( abs `  ( t `  z
) ) ) }
24 cxr 9075 . . . 4  class  RR*
25 clt 9076 . . . 4  class  <
2623, 24, 25csup 7403 . . 3  class  sup ( { x  |  E. z  e.  ~H  (
( normh `  z )  <_  1  /\  x  =  ( abs `  (
t `  z )
) ) } ,  RR* ,  <  )
272, 6, 26cmpt 4226 . 2  class  ( t  e.  ( CC  ^m  ~H )  |->  sup ( { x  |  E. z  e.  ~H  (
( normh `  z )  <_  1  /\  x  =  ( abs `  (
t `  z )
) ) } ,  RR* ,  <  ) )
281, 27wceq 1649 1  wff  normfn  =  ( t  e.  ( CC 
^m  ~H )  |->  sup ( { x  |  E. z  e.  ~H  (
( normh `  z )  <_  1  /\  x  =  ( abs `  (
t `  z )
) ) } ,  RR* ,  <  ) )
Colors of variables: wff set class
This definition is referenced by:  nmfnval  23332
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