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Definition df-nmhm 18235
Description: Define a normed module homomorphism, also known as a bounded linear operator. This is a module homomorphism (a linear function) such that the operator norm is finite, or equivalently there is a constant  c such that (Contributed by Mario Carneiro, 18-Oct-2015.)
Assertion
Ref Expression
df-nmhm  |- NMHom  =  ( s  e. NrmMod ,  t  e. NrmMod  |->  ( ( s LMHom  t
)  i^i  ( s NGHom  t ) ) )
Distinct variable group:    t, s

Detailed syntax breakdown of Definition df-nmhm
StepHypRef Expression
1 cnmhm 18232 . 2  class NMHom
2 vs . . 3  set  s
3 vt . . 3  set  t
4 cnlm 18119 . . 3  class NrmMod
52cv 1631 . . . . 5  class  s
63cv 1631 . . . . 5  class  t
7 clmhm 15792 . . . . 5  class LMHom
85, 6, 7co 5874 . . . 4  class  ( s LMHom 
t )
9 cnghm 18231 . . . . 5  class NGHom
105, 6, 9co 5874 . . . 4  class  ( s NGHom 
t )
118, 10cin 3164 . . 3  class  ( ( s LMHom  t )  i^i  ( s NGHom  t ) )
122, 3, 4, 4, 11cmpt2 5876 . 2  class  ( s  e. NrmMod ,  t  e. NrmMod  |->  ( ( s LMHom  t
)  i^i  ( s NGHom  t ) ) )
131, 12wceq 1632 1  wff NMHom  =  ( s  e. NrmMod ,  t  e. NrmMod  |->  ( ( s LMHom  t
)  i^i  ( s NGHom  t ) ) )
Colors of variables: wff set class
This definition is referenced by:  reldmnmhm  18238  isnmhm  18271
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