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Definition df-hfsum 22313
Description: Define the sum of two Hilbert space functionals. Definition of [Beran] p. 111. Note that unlike some authors, we define a functional as any function from  ~H to  CC, not just linear (or bounded linear) ones. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
Assertion
Ref Expression
df-hfsum  |-  +fn  =  ( f  e.  ( CC  ^m  ~H ) ,  g  e.  ( CC  ^m  ~H )  |->  ( x  e.  ~H  |->  ( ( f `  x
)  +  ( g `
 x ) ) ) )
Distinct variable group:    f, g, x

Detailed syntax breakdown of Definition df-hfsum
StepHypRef Expression
1 chfs 21521 . 2  class  +fn
2 vf . . 3  set  f
3 vg . . 3  set  g
4 cc 8735 . . . 4  class  CC
5 chil 21499 . . . 4  class  ~H
6 cmap 6772 . . . 4  class  ^m
74, 5, 6co 5858 . . 3  class  ( CC 
^m  ~H )
8 vx . . . 4  set  x
98cv 1622 . . . . . 6  class  x
102cv 1622 . . . . . 6  class  f
119, 10cfv 5255 . . . . 5  class  ( f `
 x )
123cv 1622 . . . . . 6  class  g
139, 12cfv 5255 . . . . 5  class  ( g `
 x )
14 caddc 8740 . . . . 5  class  +
1511, 13, 14co 5858 . . . 4  class  ( ( f `  x )  +  ( g `  x ) )
168, 5, 15cmpt 4077 . . 3  class  ( x  e.  ~H  |->  ( ( f `  x )  +  ( g `  x ) ) )
172, 3, 7, 7, 16cmpt2 5860 . 2  class  ( f  e.  ( CC  ^m  ~H ) ,  g  e.  ( CC  ^m  ~H )  |->  ( x  e. 
~H  |->  ( ( f `
 x )  +  ( g `  x
) ) ) )
181, 17wceq 1623 1  wff  +fn  =  ( f  e.  ( CC  ^m  ~H ) ,  g  e.  ( CC  ^m  ~H )  |->  ( x  e.  ~H  |->  ( ( f `  x
)  +  ( g `
 x ) ) ) )
Colors of variables: wff set class
This definition is referenced by:  hfsmval  22318
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