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Definition df-hfmul 22330
Description: Define the scalar product with a Hilbert space functional. Definition of [Beran] p. 111. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
Assertion
Ref Expression
df-hfmul  |-  .fn  =  ( f  e.  CC ,  g  e.  ( CC  ^m  ~H )  |->  ( x  e.  ~H  |->  ( f  x.  ( g `
 x ) ) ) )
Distinct variable group:    f, g, x

Detailed syntax breakdown of Definition df-hfmul
StepHypRef Expression
1 chft 21538 . 2  class  .fn
2 vf . . 3  set  f
3 vg . . 3  set  g
4 cc 8751 . . 3  class  CC
5 chil 21515 . . . 4  class  ~H
6 cmap 6788 . . . 4  class  ^m
74, 5, 6co 5874 . . 3  class  ( CC 
^m  ~H )
8 vx . . . 4  set  x
92cv 1631 . . . . 5  class  f
108cv 1631 . . . . . 6  class  x
113cv 1631 . . . . . 6  class  g
1210, 11cfv 5271 . . . . 5  class  ( g `
 x )
13 cmul 8758 . . . . 5  class  x.
149, 12, 13co 5874 . . . 4  class  ( f  x.  ( g `  x ) )
158, 5, 14cmpt 4093 . . 3  class  ( x  e.  ~H  |->  ( f  x.  ( g `  x ) ) )
162, 3, 4, 7, 15cmpt2 5876 . 2  class  ( f  e.  CC ,  g  e.  ( CC  ^m  ~H )  |->  ( x  e.  ~H  |->  ( f  x.  ( g `  x ) ) ) )
171, 16wceq 1632 1  wff  .fn  =  ( f  e.  CC ,  g  e.  ( CC  ^m  ~H )  |->  ( x  e.  ~H  |->  ( f  x.  ( g `
 x ) ) ) )
Colors of variables: wff set class
This definition is referenced by:  hfmmval  22335
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