:: CLOPBAN1 semantic presentation
theorem Th1: :: CLOPBAN1:1
definition
let c
1 be non
empty set ;
let c
2 be
ComplexLinearSpace;
func FuncExtMult c
1,c
2 -> Function of
[:COMPLEX ,(Funcs a1,the carrier of a2):],
Funcs a
1,the
carrier of a
2 means :
Def1:
:: CLOPBAN1:def 1
for b
1 being
Complexfor b
2 being
Element of
Funcs a
1,the
carrier of a
2for b
3 being
Element of a
1 holds
(a3 . [b1,b2]) . b
3 = b
1 * (b2 . b3);
existence
ex b1 being Function of [:COMPLEX ,(Funcs c1,the carrier of c2):], Funcs c1,the carrier of c2 st
for b2 being Complex
for b3 being Element of Funcs c1,the carrier of c2
for b4 being Element of c1 holds (b1 . [b2,b3]) . b4 = b2 * (b3 . b4)
by Th1;
uniqueness
for b1, b2 being Function of [:COMPLEX ,(Funcs c1,the carrier of c2):], Funcs c1,the carrier of c2 holds
( ( for b3 being Complex
for b4 being Element of Funcs c1,the carrier of c2
for b5 being Element of c1 holds (b1 . [b3,b4]) . b5 = b3 * (b4 . b5) ) & ( for b3 being Complex
for b4 being Element of Funcs c1,the carrier of c2
for b5 being Element of c1 holds (b2 . [b3,b4]) . b5 = b3 * (b4 . b5) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines FuncExtMult CLOPBAN1:def 1 :
theorem Th2: :: CLOPBAN1:2
theorem Th3: :: CLOPBAN1:3
theorem Th4: :: CLOPBAN1:4
theorem Th5: :: CLOPBAN1:5
for b
1 being non
empty set for b
2 being
ComplexLinearSpacefor b
3, b
4, b
5 being
Element of
Funcs b
1,the
carrier of b
2 holds
(FuncAdd b1,b2) . b
3,
((FuncAdd b1,b2) . b4,b5) = (FuncAdd b1,b2) . ((FuncAdd b1,b2) . b3,b4),b
5
theorem Th6: :: CLOPBAN1:6
theorem Th7: :: CLOPBAN1:7
theorem Th8: :: CLOPBAN1:8
theorem Th9: :: CLOPBAN1:9
theorem Th10: :: CLOPBAN1:10
for b
1 being non
empty set for b
2 being
ComplexLinearSpacefor b
3 being
Element of
Funcs b
1,the
carrier of b
2for b
4, b
5 being
Complex holds
(FuncAdd b1,b2) . ((FuncExtMult b1,b2) . [b4,b3]),
((FuncExtMult b1,b2) . [b5,b3]) = (FuncExtMult b1,b2) . [(b4 + b5),b3]
Lemma12:
for b1 being non empty set
for b2 being ComplexLinearSpace
for b3, b4 being Element of Funcs b1,the carrier of b2
for b5 being Complex holds (FuncAdd b1,b2) . ((FuncExtMult b1,b2) . [b5,b3]),((FuncExtMult b1,b2) . [b5,b4]) = (FuncExtMult b1,b2) . [b5,((FuncAdd b1,b2) . b3,b4)]
theorem Th11: :: CLOPBAN1:11
definition
let c
1 be non
empty set ;
let c
2 be
ComplexLinearSpace;
func ComplexVectSpace c
1,c
2 -> ComplexLinearSpace equals :: CLOPBAN1:def 2
CLSStruct(#
(Funcs a1,the carrier of a2),
(FuncZero a1,a2),
(FuncAdd a1,a2),
(FuncExtMult a1,a2) #);
coherence
CLSStruct(# (Funcs c1,the carrier of c2),(FuncZero c1,c2),(FuncAdd c1,c2),(FuncExtMult c1,c2) #) is ComplexLinearSpace
by Th11;
end;
:: deftheorem Def2 defines ComplexVectSpace CLOPBAN1:def 2 :
theorem Th12: :: CLOPBAN1:12
theorem Th13: :: CLOPBAN1:13
theorem Th14: :: CLOPBAN1:14
:: deftheorem Def3 defines additive CLOPBAN1:def 3 :
:: deftheorem Def4 defines homogeneous CLOPBAN1:def 4 :
definition
let c
1, c
2 be
ComplexLinearSpace;
func LinearOperators c
1,c
2 -> Subset of
(ComplexVectSpace the carrier of a1,a2) means :
Def5:
:: CLOPBAN1:def 5
for b
1 being
set holds
( b
1 in a
3 iff b
1 is
LinearOperator of a
1,a
2 );
existence
ex b1 being Subset of (ComplexVectSpace the carrier of c1,c2) st
for b2 being set holds
( b2 in b1 iff b2 is LinearOperator of c1,c2 )
uniqueness
for b1, b2 being Subset of (ComplexVectSpace the carrier of c1,c2) holds
( ( for b3 being set holds
( b3 in b1 iff b3 is LinearOperator of c1,c2 ) ) & ( for b3 being set holds
( b3 in b2 iff b3 is LinearOperator of c1,c2 ) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines LinearOperators CLOPBAN1:def 5 :
theorem Th15: :: CLOPBAN1:15
theorem Th16: :: CLOPBAN1:16
for b
1, b
2 being
ComplexLinearSpace holds
CLSStruct(#
(LinearOperators b1,b2),
(Zero_ (LinearOperators b1,b2),(ComplexVectSpace the carrier of b1,b2)),
(Add_ (LinearOperators b1,b2),(ComplexVectSpace the carrier of b1,b2)),
(Mult_ (LinearOperators b1,b2),(ComplexVectSpace the carrier of b1,b2)) #) is
Subspace of
ComplexVectSpace the
carrier of b
1,b
2
registration
let c
1, c
2 be
ComplexLinearSpace;
cluster CLSStruct(#
(LinearOperators a1,a2),
(Zero_ (LinearOperators a1,a2),(ComplexVectSpace the carrier of a1,a2)),
(Add_ (LinearOperators a1,a2),(ComplexVectSpace the carrier of a1,a2)),
(Mult_ (LinearOperators a1,a2),(ComplexVectSpace the carrier of a1,a2)) #)
-> Abelian add-associative right_zeroed right_complementable ComplexLinearSpace-like ;
coherence
( CLSStruct(# (LinearOperators c1,c2),(Zero_ (LinearOperators c1,c2),(ComplexVectSpace the carrier of c1,c2)),(Add_ (LinearOperators c1,c2),(ComplexVectSpace the carrier of c1,c2)),(Mult_ (LinearOperators c1,c2),(ComplexVectSpace the carrier of c1,c2)) #) is Abelian & CLSStruct(# (LinearOperators c1,c2),(Zero_ (LinearOperators c1,c2),(ComplexVectSpace the carrier of c1,c2)),(Add_ (LinearOperators c1,c2),(ComplexVectSpace the carrier of c1,c2)),(Mult_ (LinearOperators c1,c2),(ComplexVectSpace the carrier of c1,c2)) #) is add-associative & CLSStruct(# (LinearOperators c1,c2),(Zero_ (LinearOperators c1,c2),(ComplexVectSpace the carrier of c1,c2)),(Add_ (LinearOperators c1,c2),(ComplexVectSpace the carrier of c1,c2)),(Mult_ (LinearOperators c1,c2),(ComplexVectSpace the carrier of c1,c2)) #) is right_zeroed & CLSStruct(# (LinearOperators c1,c2),(Zero_ (LinearOperators c1,c2),(ComplexVectSpace the carrier of c1,c2)),(Add_ (LinearOperators c1,c2),(ComplexVectSpace the carrier of c1,c2)),(Mult_ (LinearOperators c1,c2),(ComplexVectSpace the carrier of c1,c2)) #) is right_complementable & CLSStruct(# (LinearOperators c1,c2),(Zero_ (LinearOperators c1,c2),(ComplexVectSpace the carrier of c1,c2)),(Add_ (LinearOperators c1,c2),(ComplexVectSpace the carrier of c1,c2)),(Mult_ (LinearOperators c1,c2),(ComplexVectSpace the carrier of c1,c2)) #) is ComplexLinearSpace-like )
by Th16;
end;
definition
let c
1, c
2 be
ComplexLinearSpace;
func C_VectorSpace_of_LinearOperators c
1,c
2 -> ComplexLinearSpace equals :: CLOPBAN1:def 6
CLSStruct(#
(LinearOperators a1,a2),
(Zero_ (LinearOperators a1,a2),(ComplexVectSpace the carrier of a1,a2)),
(Add_ (LinearOperators a1,a2),(ComplexVectSpace the carrier of a1,a2)),
(Mult_ (LinearOperators a1,a2),(ComplexVectSpace the carrier of a1,a2)) #);
coherence
CLSStruct(# (LinearOperators c1,c2),(Zero_ (LinearOperators c1,c2),(ComplexVectSpace the carrier of c1,c2)),(Add_ (LinearOperators c1,c2),(ComplexVectSpace the carrier of c1,c2)),(Mult_ (LinearOperators c1,c2),(ComplexVectSpace the carrier of c1,c2)) #) is ComplexLinearSpace
;
end;
:: deftheorem Def6 defines C_VectorSpace_of_LinearOperators CLOPBAN1:def 6 :
for b
1, b
2 being
ComplexLinearSpace holds
C_VectorSpace_of_LinearOperators b
1,b
2 = CLSStruct(#
(LinearOperators b1,b2),
(Zero_ (LinearOperators b1,b2),(ComplexVectSpace the carrier of b1,b2)),
(Add_ (LinearOperators b1,b2),(ComplexVectSpace the carrier of b1,b2)),
(Mult_ (LinearOperators b1,b2),(ComplexVectSpace the carrier of b1,b2)) #);
theorem Th17: :: CLOPBAN1:17
canceled;
theorem Th18: :: CLOPBAN1:18
theorem Th19: :: CLOPBAN1:19
theorem Th20: :: CLOPBAN1:20
theorem Th21: :: CLOPBAN1:21
theorem Th22: :: CLOPBAN1:22
:: deftheorem Def7 defines bounded CLOPBAN1:def 7 :
theorem Th23: :: CLOPBAN1:23
definition
let c
1, c
2 be
ComplexNormSpace;
func BoundedLinearOperators c
1,c
2 -> Subset of
(C_VectorSpace_of_LinearOperators a1,a2) means :
Def8:
:: CLOPBAN1:def 8
for b
1 being
set holds
( b
1 in a
3 iff b
1 is
bounded LinearOperator of a
1,a
2 );
existence
ex b1 being Subset of (C_VectorSpace_of_LinearOperators c1,c2) st
for b2 being set holds
( b2 in b1 iff b2 is bounded LinearOperator of c1,c2 )
uniqueness
for b1, b2 being Subset of (C_VectorSpace_of_LinearOperators c1,c2) holds
( ( for b3 being set holds
( b3 in b1 iff b3 is bounded LinearOperator of c1,c2 ) ) & ( for b3 being set holds
( b3 in b2 iff b3 is bounded LinearOperator of c1,c2 ) ) implies b1 = b2 )
end;
:: deftheorem Def8 defines BoundedLinearOperators CLOPBAN1:def 8 :
theorem Th24: :: CLOPBAN1:24
theorem Th25: :: CLOPBAN1:25
for b
1, b
2 being
ComplexNormSpace holds
CLSStruct(#
(BoundedLinearOperators b1,b2),
(Zero_ (BoundedLinearOperators b1,b2),(C_VectorSpace_of_LinearOperators b1,b2)),
(Add_ (BoundedLinearOperators b1,b2),(C_VectorSpace_of_LinearOperators b1,b2)),
(Mult_ (BoundedLinearOperators b1,b2),(C_VectorSpace_of_LinearOperators b1,b2)) #) is
Subspace of
C_VectorSpace_of_LinearOperators b
1,b
2
registration
let c
1, c
2 be
ComplexNormSpace;
cluster CLSStruct(#
(BoundedLinearOperators a1,a2),
(Zero_ (BoundedLinearOperators a1,a2),(C_VectorSpace_of_LinearOperators a1,a2)),
(Add_ (BoundedLinearOperators a1,a2),(C_VectorSpace_of_LinearOperators a1,a2)),
(Mult_ (BoundedLinearOperators a1,a2),(C_VectorSpace_of_LinearOperators a1,a2)) #)
-> Abelian add-associative right_zeroed right_complementable ComplexLinearSpace-like ;
coherence
( CLSStruct(# (BoundedLinearOperators c1,c2),(Zero_ (BoundedLinearOperators c1,c2),(C_VectorSpace_of_LinearOperators c1,c2)),(Add_ (BoundedLinearOperators c1,c2),(C_VectorSpace_of_LinearOperators c1,c2)),(Mult_ (BoundedLinearOperators c1,c2),(C_VectorSpace_of_LinearOperators c1,c2)) #) is Abelian & CLSStruct(# (BoundedLinearOperators c1,c2),(Zero_ (BoundedLinearOperators c1,c2),(C_VectorSpace_of_LinearOperators c1,c2)),(Add_ (BoundedLinearOperators c1,c2),(C_VectorSpace_of_LinearOperators c1,c2)),(Mult_ (BoundedLinearOperators c1,c2),(C_VectorSpace_of_LinearOperators c1,c2)) #) is add-associative & CLSStruct(# (BoundedLinearOperators c1,c2),(Zero_ (BoundedLinearOperators c1,c2),(C_VectorSpace_of_LinearOperators c1,c2)),(Add_ (BoundedLinearOperators c1,c2),(C_VectorSpace_of_LinearOperators c1,c2)),(Mult_ (BoundedLinearOperators c1,c2),(C_VectorSpace_of_LinearOperators c1,c2)) #) is right_zeroed & CLSStruct(# (BoundedLinearOperators c1,c2),(Zero_ (BoundedLinearOperators c1,c2),(C_VectorSpace_of_LinearOperators c1,c2)),(Add_ (BoundedLinearOperators c1,c2),(C_VectorSpace_of_LinearOperators c1,c2)),(Mult_ (BoundedLinearOperators c1,c2),(C_VectorSpace_of_LinearOperators c1,c2)) #) is right_complementable & CLSStruct(# (BoundedLinearOperators c1,c2),(Zero_ (BoundedLinearOperators c1,c2),(C_VectorSpace_of_LinearOperators c1,c2)),(Add_ (BoundedLinearOperators c1,c2),(C_VectorSpace_of_LinearOperators c1,c2)),(Mult_ (BoundedLinearOperators c1,c2),(C_VectorSpace_of_LinearOperators c1,c2)) #) is ComplexLinearSpace-like )
by Th25;
end;
definition
let c
1, c
2 be
ComplexNormSpace;
func C_VectorSpace_of_BoundedLinearOperators c
1,c
2 -> ComplexLinearSpace equals :: CLOPBAN1:def 9
CLSStruct(#
(BoundedLinearOperators a1,a2),
(Zero_ (BoundedLinearOperators a1,a2),(C_VectorSpace_of_LinearOperators a1,a2)),
(Add_ (BoundedLinearOperators a1,a2),(C_VectorSpace_of_LinearOperators a1,a2)),
(Mult_ (BoundedLinearOperators a1,a2),(C_VectorSpace_of_LinearOperators a1,a2)) #);
coherence
CLSStruct(# (BoundedLinearOperators c1,c2),(Zero_ (BoundedLinearOperators c1,c2),(C_VectorSpace_of_LinearOperators c1,c2)),(Add_ (BoundedLinearOperators c1,c2),(C_VectorSpace_of_LinearOperators c1,c2)),(Mult_ (BoundedLinearOperators c1,c2),(C_VectorSpace_of_LinearOperators c1,c2)) #) is ComplexLinearSpace
;
end;
:: deftheorem Def9 defines C_VectorSpace_of_BoundedLinearOperators CLOPBAN1:def 9 :
for b
1, b
2 being
ComplexNormSpace holds
C_VectorSpace_of_BoundedLinearOperators b
1,b
2 = CLSStruct(#
(BoundedLinearOperators b1,b2),
(Zero_ (BoundedLinearOperators b1,b2),(C_VectorSpace_of_LinearOperators b1,b2)),
(Add_ (BoundedLinearOperators b1,b2),(C_VectorSpace_of_LinearOperators b1,b2)),
(Mult_ (BoundedLinearOperators b1,b2),(C_VectorSpace_of_LinearOperators b1,b2)) #);
theorem Th26: :: CLOPBAN1:26
canceled;
theorem Th27: :: CLOPBAN1:27
theorem Th28: :: CLOPBAN1:28
theorem Th29: :: CLOPBAN1:29
:: deftheorem Def10 defines modetrans CLOPBAN1:def 10 :
:: deftheorem Def11 defines PreNorms CLOPBAN1:def 11 :
theorem Th30: :: CLOPBAN1:30
theorem Th31: :: CLOPBAN1:31
theorem Th32: :: CLOPBAN1:32
definition
let c
1, c
2 be
ComplexNormSpace;
func BoundedLinearOperatorsNorm c
1,c
2 -> Function of
BoundedLinearOperators a
1,a
2,
REAL means :
Def12:
:: CLOPBAN1:def 12
for b
1 being
set holds
( b
1 in BoundedLinearOperators a
1,a
2 implies a
3 . b
1 = sup (PreNorms (modetrans b1,a1,a2)) );
existence
ex b1 being Function of BoundedLinearOperators c1,c2, REAL st
for b2 being set holds
( b2 in BoundedLinearOperators c1,c2 implies b1 . b2 = sup (PreNorms (modetrans b2,c1,c2)) )
by Th32;
uniqueness
for b1, b2 being Function of BoundedLinearOperators c1,c2, REAL holds
( ( for b3 being set holds
( b3 in BoundedLinearOperators c1,c2 implies b1 . b3 = sup (PreNorms (modetrans b3,c1,c2)) ) ) & ( for b3 being set holds
( b3 in BoundedLinearOperators c1,c2 implies b2 . b3 = sup (PreNorms (modetrans b3,c1,c2)) ) ) implies b1 = b2 )
end;
:: deftheorem Def12 defines BoundedLinearOperatorsNorm CLOPBAN1:def 12 :
theorem Th33: :: CLOPBAN1:33
theorem Th34: :: CLOPBAN1:34
definition
let c
1, c
2 be
ComplexNormSpace;
func C_NormSpace_of_BoundedLinearOperators c
1,c
2 -> non
empty CNORMSTR equals :: CLOPBAN1:def 13
CNORMSTR(#
(BoundedLinearOperators a1,a2),
(Zero_ (BoundedLinearOperators a1,a2),(C_VectorSpace_of_LinearOperators a1,a2)),
(Add_ (BoundedLinearOperators a1,a2),(C_VectorSpace_of_LinearOperators a1,a2)),
(Mult_ (BoundedLinearOperators a1,a2),(C_VectorSpace_of_LinearOperators a1,a2)),
(BoundedLinearOperatorsNorm a1,a2) #);
coherence
CNORMSTR(# (BoundedLinearOperators c1,c2),(Zero_ (BoundedLinearOperators c1,c2),(C_VectorSpace_of_LinearOperators c1,c2)),(Add_ (BoundedLinearOperators c1,c2),(C_VectorSpace_of_LinearOperators c1,c2)),(Mult_ (BoundedLinearOperators c1,c2),(C_VectorSpace_of_LinearOperators c1,c2)),(BoundedLinearOperatorsNorm c1,c2) #) is non empty CNORMSTR
;
end;
:: deftheorem Def13 defines C_NormSpace_of_BoundedLinearOperators CLOPBAN1:def 13 :
for b
1, b
2 being
ComplexNormSpace holds
C_NormSpace_of_BoundedLinearOperators b
1,b
2 = CNORMSTR(#
(BoundedLinearOperators b1,b2),
(Zero_ (BoundedLinearOperators b1,b2),(C_VectorSpace_of_LinearOperators b1,b2)),
(Add_ (BoundedLinearOperators b1,b2),(C_VectorSpace_of_LinearOperators b1,b2)),
(Mult_ (BoundedLinearOperators b1,b2),(C_VectorSpace_of_LinearOperators b1,b2)),
(BoundedLinearOperatorsNorm b1,b2) #);
theorem Th35: :: CLOPBAN1:35
theorem Th36: :: CLOPBAN1:36
theorem Th37: :: CLOPBAN1:37
theorem Th38: :: CLOPBAN1:38
theorem Th39: :: CLOPBAN1:39
theorem Th40: :: CLOPBAN1:40
theorem Th41: :: CLOPBAN1:41
theorem Th42: :: CLOPBAN1:42
theorem Th43: :: CLOPBAN1:43
theorem Th44: :: CLOPBAN1:44
:: deftheorem Def14 defines complete CLOPBAN1:def 14 :
Lemma52:
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 Th45: :: CLOPBAN1:45
theorem Th46: :: CLOPBAN1:46
theorem Th47: :: CLOPBAN1:47