:: CSSPACE4 semantic presentation
Lemma1:
for b1 being Real_Sequence
for b2 being real number holds
( ( for b3 being Nat holds b1 . b3 <= b2 ) implies sup (rng b1) <= b2 )
Lemma2:
for b1 being Real_Sequence holds
( b1 is bounded implies for b2 being Nat holds b1 . b2 <= sup (rng b1) )
:: deftheorem Def1 defines the_set_of_BoundedComplexSequences CSSPACE4:def 1 :
Lemma4:
for b1, b2 being Complex_Sequence holds
( b1 is bounded & b2 is bounded implies b1 + b2 is bounded )
Lemma5:
for b1 being Complex
for b2 being Complex_Sequence holds
( b2 is bounded implies b1 (#) b2 is bounded )
Lemma6:
CLSStruct(# the_set_of_BoundedComplexSequences ,(Zero_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Add_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Mult_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ) #) is Subspace of Linear_Space_of_ComplexSequences
by CSSPACE:13;
registration
cluster CLSStruct(#
the_set_of_BoundedComplexSequences ,
(Zero_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),
(Add_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),
(Mult_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ) #)
-> Abelian add-associative right_zeroed right_complementable ComplexLinearSpace-like ;
coherence
( CLSStruct(# the_set_of_BoundedComplexSequences ,(Zero_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Add_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Mult_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ) #) is Abelian & CLSStruct(# the_set_of_BoundedComplexSequences ,(Zero_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Add_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Mult_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ) #) is add-associative & CLSStruct(# the_set_of_BoundedComplexSequences ,(Zero_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Add_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Mult_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ) #) is right_zeroed & CLSStruct(# the_set_of_BoundedComplexSequences ,(Zero_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Add_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Mult_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ) #) is right_complementable & CLSStruct(# the_set_of_BoundedComplexSequences ,(Zero_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Add_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Mult_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ) #) is ComplexLinearSpace-like )
by CSSPACE:13;
end;
Lemma7:
( CLSStruct(# the_set_of_BoundedComplexSequences ,(Zero_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Add_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Mult_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ) #) is Abelian & CLSStruct(# the_set_of_BoundedComplexSequences ,(Zero_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Add_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Mult_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ) #) is add-associative & CLSStruct(# the_set_of_BoundedComplexSequences ,(Zero_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Add_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Mult_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ) #) is right_zeroed & CLSStruct(# the_set_of_BoundedComplexSequences ,(Zero_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Add_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Mult_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ) #) is right_complementable & CLSStruct(# the_set_of_BoundedComplexSequences ,(Zero_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Add_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Mult_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ) #) is ComplexLinearSpace-like )
;
Lemma8:
ex b1 being Function of the_set_of_BoundedComplexSequences , REAL st
for b2 being set holds
( b2 in the_set_of_BoundedComplexSequences implies b1 . b2 = sup (rng |.(seq_id b2).|) )
:: deftheorem Def2 defines Complex_linfty_norm CSSPACE4:def 2 :
Lemma10:
for b1 being Complex_Sequence holds
( ( for b2 being Nat holds b1 . b2 = 0c ) implies ( b1 is bounded & sup (rng |.b1.|) = 0 ) )
Lemma11:
for b1 being Complex_Sequence holds
( b1 is bounded implies |.b1.| is bounded )
Lemma12:
for b1 being Complex_Sequence holds
( |.b1.| is bounded implies b1 is bounded )
Lemma13:
for b1 being Complex_Sequence holds
( b1 is bounded & sup (rng |.b1.|) = 0 implies for b2 being Nat holds b1 . b2 = 0c )
theorem Th1: :: CSSPACE4:1
canceled;
theorem Th2: :: CSSPACE4:2
registration
cluster CNORMSTR(#
the_set_of_BoundedComplexSequences ,
(Zero_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),
(Add_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),
(Mult_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),
Complex_linfty_norm #)
-> Abelian add-associative right_zeroed right_complementable ComplexLinearSpace-like ;
coherence
( CNORMSTR(# the_set_of_BoundedComplexSequences ,(Zero_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Add_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Mult_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),Complex_linfty_norm #) is Abelian & CNORMSTR(# the_set_of_BoundedComplexSequences ,(Zero_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Add_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Mult_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),Complex_linfty_norm #) is add-associative & CNORMSTR(# the_set_of_BoundedComplexSequences ,(Zero_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Add_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Mult_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),Complex_linfty_norm #) is right_zeroed & CNORMSTR(# the_set_of_BoundedComplexSequences ,(Zero_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Add_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Mult_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),Complex_linfty_norm #) is right_complementable & CNORMSTR(# the_set_of_BoundedComplexSequences ,(Zero_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Add_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Mult_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),Complex_linfty_norm #) is ComplexLinearSpace-like )
by Lemma7, CSSPACE3:4;
end;
definition
func Complex_linfty_Space -> non
empty CNORMSTR equals :: CSSPACE4:def 3
CNORMSTR(#
the_set_of_BoundedComplexSequences ,
(Zero_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),
(Add_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),
(Mult_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),
Complex_linfty_norm #);
coherence
CNORMSTR(# the_set_of_BoundedComplexSequences ,(Zero_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Add_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),(Mult_ the_set_of_BoundedComplexSequences ,Linear_Space_of_ComplexSequences ),Complex_linfty_norm #) is non empty CNORMSTR
;
end;
:: deftheorem Def3 defines Complex_linfty_Space CSSPACE4:def 3 :
theorem Th3: :: CSSPACE4:3
theorem Th4: :: CSSPACE4:4
Lemma16:
for b1, b2, b3 being Complex_Sequence holds
( b1 = b2 - b3 iff for b4 being Nat holds b1 . b4 = (b2 . b4) - (b3 . b4) )
theorem Th5: :: CSSPACE4:5
theorem Th6: :: CSSPACE4:6
:: deftheorem Def4 defines bounded CSSPACE4:def 4 :
theorem Th7: :: CSSPACE4:7
:: deftheorem Def5 defines ComplexBoundedFunctions CSSPACE4:def 5 :
theorem Th8: :: CSSPACE4:8
theorem Th9: :: CSSPACE4:9
for b
1 being non
empty set for b
2 being
ComplexNormSpace holds
CLSStruct(#
(ComplexBoundedFunctions b1,b2),
(Zero_ (ComplexBoundedFunctions b1,b2),(ComplexVectSpace b1,b2)),
(Add_ (ComplexBoundedFunctions b1,b2),(ComplexVectSpace b1,b2)),
(Mult_ (ComplexBoundedFunctions b1,b2),(ComplexVectSpace b1,b2)) #) is
Subspace of
ComplexVectSpace b
1,b
2
registration
let c
1 be non
empty set ;
let c
2 be
ComplexNormSpace;
cluster CLSStruct(#
(ComplexBoundedFunctions a1,a2),
(Zero_ (ComplexBoundedFunctions a1,a2),(ComplexVectSpace a1,a2)),
(Add_ (ComplexBoundedFunctions a1,a2),(ComplexVectSpace a1,a2)),
(Mult_ (ComplexBoundedFunctions a1,a2),(ComplexVectSpace a1,a2)) #)
-> Abelian add-associative right_zeroed right_complementable ComplexLinearSpace-like ;
coherence
( CLSStruct(# (ComplexBoundedFunctions c1,c2),(Zero_ (ComplexBoundedFunctions c1,c2),(ComplexVectSpace c1,c2)),(Add_ (ComplexBoundedFunctions c1,c2),(ComplexVectSpace c1,c2)),(Mult_ (ComplexBoundedFunctions c1,c2),(ComplexVectSpace c1,c2)) #) is Abelian & CLSStruct(# (ComplexBoundedFunctions c1,c2),(Zero_ (ComplexBoundedFunctions c1,c2),(ComplexVectSpace c1,c2)),(Add_ (ComplexBoundedFunctions c1,c2),(ComplexVectSpace c1,c2)),(Mult_ (ComplexBoundedFunctions c1,c2),(ComplexVectSpace c1,c2)) #) is add-associative & CLSStruct(# (ComplexBoundedFunctions c1,c2),(Zero_ (ComplexBoundedFunctions c1,c2),(ComplexVectSpace c1,c2)),(Add_ (ComplexBoundedFunctions c1,c2),(ComplexVectSpace c1,c2)),(Mult_ (ComplexBoundedFunctions c1,c2),(ComplexVectSpace c1,c2)) #) is right_zeroed & CLSStruct(# (ComplexBoundedFunctions c1,c2),(Zero_ (ComplexBoundedFunctions c1,c2),(ComplexVectSpace c1,c2)),(Add_ (ComplexBoundedFunctions c1,c2),(ComplexVectSpace c1,c2)),(Mult_ (ComplexBoundedFunctions c1,c2),(ComplexVectSpace c1,c2)) #) is right_complementable & CLSStruct(# (ComplexBoundedFunctions c1,c2),(Zero_ (ComplexBoundedFunctions c1,c2),(ComplexVectSpace c1,c2)),(Add_ (ComplexBoundedFunctions c1,c2),(ComplexVectSpace c1,c2)),(Mult_ (ComplexBoundedFunctions c1,c2),(ComplexVectSpace c1,c2)) #) is ComplexLinearSpace-like )
by Th9;
end;
definition
let c
1 be non
empty set ;
let c
2 be
ComplexNormSpace;
func C_VectorSpace_of_BoundedFunctions c
1,c
2 -> ComplexLinearSpace equals :: CSSPACE4:def 6
CLSStruct(#
(ComplexBoundedFunctions a1,a2),
(Zero_ (ComplexBoundedFunctions a1,a2),(ComplexVectSpace a1,a2)),
(Add_ (ComplexBoundedFunctions a1,a2),(ComplexVectSpace a1,a2)),
(Mult_ (ComplexBoundedFunctions a1,a2),(ComplexVectSpace a1,a2)) #);
coherence
CLSStruct(# (ComplexBoundedFunctions c1,c2),(Zero_ (ComplexBoundedFunctions c1,c2),(ComplexVectSpace c1,c2)),(Add_ (ComplexBoundedFunctions c1,c2),(ComplexVectSpace c1,c2)),(Mult_ (ComplexBoundedFunctions c1,c2),(ComplexVectSpace c1,c2)) #) is ComplexLinearSpace
;
end;
:: deftheorem Def6 defines C_VectorSpace_of_BoundedFunctions CSSPACE4:def 6 :
for b
1 being non
empty set for b
2 being
ComplexNormSpace holds
C_VectorSpace_of_BoundedFunctions b
1,b
2 = CLSStruct(#
(ComplexBoundedFunctions b1,b2),
(Zero_ (ComplexBoundedFunctions b1,b2),(ComplexVectSpace b1,b2)),
(Add_ (ComplexBoundedFunctions b1,b2),(ComplexVectSpace b1,b2)),
(Mult_ (ComplexBoundedFunctions b1,b2),(ComplexVectSpace b1,b2)) #);
theorem Th10: :: CSSPACE4:10
canceled;
theorem Th11: :: CSSPACE4:11
theorem Th12: :: CSSPACE4:12
theorem Th13: :: CSSPACE4:13
:: deftheorem Def7 defines modetrans CSSPACE4:def 7 :
:: deftheorem Def8 defines PreNorms CSSPACE4:def 8 :
theorem Th14: :: CSSPACE4:14
theorem Th15: :: CSSPACE4:15
theorem Th16: :: CSSPACE4:16
definition
let c
1 be non
empty set ;
let c
2 be
ComplexNormSpace;
func ComplexBoundedFunctionsNorm c
1,c
2 -> Function of
ComplexBoundedFunctions a
1,a
2,
REAL means :
Def9:
:: CSSPACE4:def 9
for b
1 being
set holds
( b
1 in ComplexBoundedFunctions a
1,a
2 implies a
3 . b
1 = sup (PreNorms (modetrans b1,a1,a2)) );
existence
ex b1 being Function of ComplexBoundedFunctions c1,c2, REAL st
for b2 being set holds
( b2 in ComplexBoundedFunctions c1,c2 implies b1 . b2 = sup (PreNorms (modetrans b2,c1,c2)) )
by Th16;
uniqueness
for b1, b2 being Function of ComplexBoundedFunctions c1,c2, REAL holds
( ( for b3 being set holds
( b3 in ComplexBoundedFunctions c1,c2 implies b1 . b3 = sup (PreNorms (modetrans b3,c1,c2)) ) ) & ( for b3 being set holds
( b3 in ComplexBoundedFunctions c1,c2 implies b2 . b3 = sup (PreNorms (modetrans b3,c1,c2)) ) ) implies b1 = b2 )
end;
:: deftheorem Def9 defines ComplexBoundedFunctionsNorm CSSPACE4:def 9 :
theorem Th17: :: CSSPACE4:17
theorem Th18: :: CSSPACE4:18
definition
let c
1 be non
empty set ;
let c
2 be
ComplexNormSpace;
func C_NormSpace_of_BoundedFunctions c
1,c
2 -> non
empty CNORMSTR equals :: CSSPACE4:def 10
CNORMSTR(#
(ComplexBoundedFunctions a1,a2),
(Zero_ (ComplexBoundedFunctions a1,a2),(ComplexVectSpace a1,a2)),
(Add_ (ComplexBoundedFunctions a1,a2),(ComplexVectSpace a1,a2)),
(Mult_ (ComplexBoundedFunctions a1,a2),(ComplexVectSpace a1,a2)),
(ComplexBoundedFunctionsNorm a1,a2) #);
coherence
CNORMSTR(# (ComplexBoundedFunctions c1,c2),(Zero_ (ComplexBoundedFunctions c1,c2),(ComplexVectSpace c1,c2)),(Add_ (ComplexBoundedFunctions c1,c2),(ComplexVectSpace c1,c2)),(Mult_ (ComplexBoundedFunctions c1,c2),(ComplexVectSpace c1,c2)),(ComplexBoundedFunctionsNorm c1,c2) #) is non empty CNORMSTR
;
end;
:: deftheorem Def10 defines C_NormSpace_of_BoundedFunctions CSSPACE4:def 10 :
for b
1 being non
empty set for b
2 being
ComplexNormSpace holds
C_NormSpace_of_BoundedFunctions b
1,b
2 = CNORMSTR(#
(ComplexBoundedFunctions b1,b2),
(Zero_ (ComplexBoundedFunctions b1,b2),(ComplexVectSpace b1,b2)),
(Add_ (ComplexBoundedFunctions b1,b2),(ComplexVectSpace b1,b2)),
(Mult_ (ComplexBoundedFunctions b1,b2),(ComplexVectSpace b1,b2)),
(ComplexBoundedFunctionsNorm b1,b2) #);
theorem Th19: :: CSSPACE4:19
theorem Th20: :: CSSPACE4:20
theorem Th21: :: CSSPACE4:21
theorem Th22: :: CSSPACE4:22
theorem Th23: :: CSSPACE4:23
theorem Th24: :: CSSPACE4:24
theorem Th25: :: CSSPACE4:25
theorem Th26: :: CSSPACE4:26
theorem Th27: :: CSSPACE4:27
theorem Th28: :: CSSPACE4:28
Lemma41:
for b1 being Real
for b2 being Real_Sequence holds
( b2 is convergent & ex b3 being Nat st
for b4 being Nat holds
( b3 <= b4 implies b2 . b4 <= b1 ) implies lim b2 <= b1 )
theorem Th29: :: CSSPACE4:29
theorem Th30: :: CSSPACE4:30
theorem Th31: :: CSSPACE4:31
theorem Th32: :: CSSPACE4:32
theorem Th33: :: CSSPACE4:33
theorem Th34: :: CSSPACE4:34