:: CSSPACE4 semantic presentation
Lemma27:
for rseq being Real_Sequence
for K being real number st ( for n being Element of NAT holds rseq . n <= K ) holds
sup (rng rseq) <= K
Lemma35:
for rseq being Real_Sequence st rseq is bounded holds
for n being Element of NAT holds rseq . n <= sup (rng rseq)
:: deftheorem Def1 defines the_set_of_BoundedComplexSequences CSSPACE4:def 1 :
Lemma42:
for seq1, seq2 being Complex_Sequence st seq1 is bounded & seq2 is bounded holds
seq1 + seq2 is bounded
Lemma48:
for c being Complex
for seq being Complex_Sequence st seq is bounded holds
c (#) seq is bounded
Lemma58:
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;
Lemma59:
( 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 )
;
Lemma60:
ex NORM being Function of the_set_of_BoundedComplexSequences , REAL st
for x being set st x in the_set_of_BoundedComplexSequences holds
NORM . x = sup (rng |.(seq_id x).|)
:: deftheorem Def2 defines Complex_linfty_norm CSSPACE4:def 2 :
Lemma68:
for seq being Complex_Sequence st ( for n being Element of NAT holds seq . n = 0c ) holds
( seq is bounded & sup (rng |.seq.|) = 0 )
Lemma70:
for seq being Complex_Sequence st seq is bounded holds
|.seq.| is bounded
Lemma71:
for seq being Complex_Sequence st |.seq.| is bounded holds
seq is bounded
Lemma72:
for seq being Complex_Sequence st seq is bounded & sup (rng |.seq.|) = 0 holds
for n being Element of NAT holds seq . n = 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 , 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
Lemma95:
for seq1, seq2, seq3 being Complex_Sequence holds
( seq1 = seq2 - seq3 iff for n being Element of NAT holds seq1 . n = (seq2 . n) - (seq3 . n) )
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
X being non
empty set for
Y being
ComplexNormSpace holds
CLSStruct(#
(ComplexBoundedFunctions X,Y),
(Zero_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),
(Add_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),
(Mult_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)) #) is
Subspace of
ComplexVectSpace X,
Y
registration
let X be non
empty set ;
let Y 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 X,Y),(Zero_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),(Add_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),(Mult_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)) #) is Abelian & CLSStruct(# (ComplexBoundedFunctions X,Y),(Zero_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),(Add_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),(Mult_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)) #) is add-associative & CLSStruct(# (ComplexBoundedFunctions X,Y),(Zero_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),(Add_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),(Mult_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)) #) is right_zeroed & CLSStruct(# (ComplexBoundedFunctions X,Y),(Zero_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),(Add_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),(Mult_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)) #) is right_complementable & CLSStruct(# (ComplexBoundedFunctions X,Y),(Zero_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),(Add_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),(Mult_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)) #) is ComplexLinearSpace-like )
by ;
end;
definition
let X be non
empty set ;
let Y be
ComplexNormSpace;
func C_VectorSpace_of_BoundedFunctions c1,
c2 -> ComplexLinearSpace equals :: CSSPACE4:def 6
CLSStruct(#
(ComplexBoundedFunctions X,Y),
(Zero_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),
(Add_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),
(Mult_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)) #);
coherence
CLSStruct(# (ComplexBoundedFunctions X,Y),(Zero_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),(Add_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),(Mult_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)) #) is ComplexLinearSpace
;
end;
:: deftheorem Def6 defines C_VectorSpace_of_BoundedFunctions CSSPACE4:def 6 :
for
X being non
empty set for
Y being
ComplexNormSpace holds
C_VectorSpace_of_BoundedFunctions X,
Y = CLSStruct(#
(ComplexBoundedFunctions X,Y),
(Zero_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),
(Add_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),
(Mult_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)) #);
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 X be non
empty set ;
let Y be
ComplexNormSpace;
func ComplexBoundedFunctionsNorm c1,
c2 -> Function of
ComplexBoundedFunctions a1,
a2,
REAL means :
Def9:
:: CSSPACE4:def 9
for
x being
set st
x in ComplexBoundedFunctions X,
Y holds
it . x = sup (PreNorms (modetrans x,X,Y));
existence
ex b1 being Function of ComplexBoundedFunctions X,Y, REAL st
for x being set st x in ComplexBoundedFunctions X,Y holds
b1 . x = sup (PreNorms (modetrans x,X,Y))
by ;
uniqueness
for b1, b2 being Function of ComplexBoundedFunctions X,Y, REAL st ( for x being set st x in ComplexBoundedFunctions X,Y holds
b1 . x = sup (PreNorms (modetrans x,X,Y)) ) & ( for x being set st x in ComplexBoundedFunctions X,Y holds
b2 . x = sup (PreNorms (modetrans x,X,Y)) ) holds
b1 = b2
end;
:: deftheorem Def9 defines ComplexBoundedFunctionsNorm CSSPACE4:def 9 :
theorem Th17: :: CSSPACE4:17
theorem Th18: :: CSSPACE4:18
definition
let X be non
empty set ;
let Y be
ComplexNormSpace;
func C_NormSpace_of_BoundedFunctions c1,
c2 -> non
empty CNORMSTR equals :: CSSPACE4:def 10
CNORMSTR(#
(ComplexBoundedFunctions X,Y),
(Zero_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),
(Add_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),
(Mult_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),
(ComplexBoundedFunctionsNorm X,Y) #);
coherence
CNORMSTR(# (ComplexBoundedFunctions X,Y),(Zero_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),(Add_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),(Mult_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),(ComplexBoundedFunctionsNorm X,Y) #) is non empty CNORMSTR
;
end;
:: deftheorem Def10 defines C_NormSpace_of_BoundedFunctions CSSPACE4:def 10 :
for
X being non
empty set for
Y being
ComplexNormSpace holds
C_NormSpace_of_BoundedFunctions X,
Y = CNORMSTR(#
(ComplexBoundedFunctions X,Y),
(Zero_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),
(Add_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),
(Mult_ (ComplexBoundedFunctions X,Y),(ComplexVectSpace X,Y)),
(ComplexBoundedFunctionsNorm X,Y) #);
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
Lemma184:
for e being Real
for seq being Real_Sequence st seq is convergent & ex k being Element of NAT st
for i being Element of NAT st k <= i holds
seq . i <= e holds
lim seq <= e
theorem Th29: :: CSSPACE4:29
theorem Th30: :: CSSPACE4:30
theorem Th31: :: CSSPACE4:31
theorem Th32: :: CSSPACE4:32
theorem Th33: :: CSSPACE4:33
theorem Th34: :: CSSPACE4:34