:: CLOPBAN2 semantic presentation
theorem Th1: :: CLOPBAN2:1
theorem Th2: :: CLOPBAN2:2
for b
1, b
2, b
3 being
ComplexNormSpacefor b
4 being
bounded LinearOperator of b
1,b
2for b
5 being
bounded LinearOperator of b
2,b
3 holds
( b
5 * b
4 is
bounded LinearOperator of b
1,b
3 & ( for b
6 being
VECTOR of b
1 holds
(
||.((b5 * b4) . b6).|| <= (((BoundedLinearOperatorsNorm b2,b3) . b5) * ((BoundedLinearOperatorsNorm b1,b2) . b4)) * ||.b6.|| &
(BoundedLinearOperatorsNorm b1,b3) . (b5 * b4) <= ((BoundedLinearOperatorsNorm b2,b3) . b5) * ((BoundedLinearOperatorsNorm b1,b2) . b4) ) ) )
definition
let c
1 be
ComplexNormSpace;
let c
2, c
3 be
Element of
BoundedLinearOperators c
1,c
1;
func c
2 + c
3 -> Element of
BoundedLinearOperators a
1,a
1 equals :: CLOPBAN2:def 1
(Add_ (BoundedLinearOperators a1,a1),(C_VectorSpace_of_LinearOperators a1,a1)) . a
2,a
3;
correctness
coherence
(Add_ (BoundedLinearOperators c1,c1),(C_VectorSpace_of_LinearOperators c1,c1)) . c2,c3 is Element of BoundedLinearOperators c1,c1;
;
end;
:: deftheorem Def1 defines + CLOPBAN2:def 1 :
definition
let c
1 be
ComplexNormSpace;
let c
2, c
3 be
Element of
BoundedLinearOperators c
1,c
1;
func c
3 * c
2 -> Element of
BoundedLinearOperators a
1,a
1 equals :: CLOPBAN2:def 2
(modetrans a3,a1,a1) * (modetrans a2,a1,a1);
correctness
coherence
(modetrans c3,c1,c1) * (modetrans c2,c1,c1) is Element of BoundedLinearOperators c1,c1;
by CLOPBAN1:def 8;
end;
:: deftheorem Def2 defines * CLOPBAN2:def 2 :
definition
let c
1 be
ComplexNormSpace;
let c
2 be
Element of
BoundedLinearOperators c
1,c
1;
let c
3 be
Complex;
func c
3 * c
2 -> Element of
BoundedLinearOperators a
1,a
1 equals :: CLOPBAN2:def 3
(Mult_ (BoundedLinearOperators a1,a1),(C_VectorSpace_of_LinearOperators a1,a1)) . a
3,a
2;
correctness
coherence
(Mult_ (BoundedLinearOperators c1,c1),(C_VectorSpace_of_LinearOperators c1,c1)) . c3,c2 is Element of BoundedLinearOperators c1,c1;
;
end;
:: deftheorem Def3 defines * CLOPBAN2:def 3 :
definition
let c
1 be
ComplexNormSpace;
func FuncMult c
1 -> BinOp of
BoundedLinearOperators a
1,a
1 means :
Def4:
:: CLOPBAN2:def 4
for b
1, b
2 being
Element of
BoundedLinearOperators a
1,a
1 holds a
2 . b
1,b
2 = b
1 * b
2;
existence
ex b1 being BinOp of BoundedLinearOperators c1,c1 st
for b2, b3 being Element of BoundedLinearOperators c1,c1 holds b1 . b2,b3 = b2 * b3
uniqueness
for b1, b2 being BinOp of BoundedLinearOperators c1,c1 holds
( ( for b3, b4 being Element of BoundedLinearOperators c1,c1 holds b1 . b3,b4 = b3 * b4 ) & ( for b3, b4 being Element of BoundedLinearOperators c1,c1 holds b2 . b3,b4 = b3 * b4 ) implies b1 = b2 )
end;
:: deftheorem Def4 defines FuncMult CLOPBAN2:def 4 :
theorem Th3: :: CLOPBAN2:3
:: deftheorem Def5 defines FuncUnit CLOPBAN2:def 5 :
theorem Th4: :: CLOPBAN2:4
theorem Th5: :: CLOPBAN2:5
theorem Th6: :: CLOPBAN2:6
theorem Th7: :: CLOPBAN2:7
theorem Th8: :: CLOPBAN2:8
theorem Th9: :: CLOPBAN2:9
theorem Th10: :: CLOPBAN2:10
theorem Th11: :: CLOPBAN2:11
theorem Th12: :: CLOPBAN2:12
definition
let c
1 be
ComplexNormSpace;
func Ring_of_BoundedLinearOperators c
1 -> doubleLoopStr equals :: CLOPBAN2:def 6
doubleLoopStr(#
(BoundedLinearOperators a1,a1),
(Add_ (BoundedLinearOperators a1,a1),(C_VectorSpace_of_LinearOperators a1,a1)),
(FuncMult a1),
(FuncUnit a1),
(Zero_ (BoundedLinearOperators a1,a1),(C_VectorSpace_of_LinearOperators a1,a1)) #);
correctness
coherence
doubleLoopStr(# (BoundedLinearOperators c1,c1),(Add_ (BoundedLinearOperators c1,c1),(C_VectorSpace_of_LinearOperators c1,c1)),(FuncMult c1),(FuncUnit c1),(Zero_ (BoundedLinearOperators c1,c1),(C_VectorSpace_of_LinearOperators c1,c1)) #) is doubleLoopStr ;
;
end;
:: deftheorem Def6 defines Ring_of_BoundedLinearOperators CLOPBAN2:def 6 :
for b
1 being
ComplexNormSpace holds
Ring_of_BoundedLinearOperators b
1 = doubleLoopStr(#
(BoundedLinearOperators b1,b1),
(Add_ (BoundedLinearOperators b1,b1),(C_VectorSpace_of_LinearOperators b1,b1)),
(FuncMult b1),
(FuncUnit b1),
(Zero_ (BoundedLinearOperators b1,b1),(C_VectorSpace_of_LinearOperators b1,b1)) #);
theorem Th13: :: CLOPBAN2:13
theorem Th14: :: CLOPBAN2:14
definition
let c
1 be
ComplexNormSpace;
func C_Algebra_of_BoundedLinearOperators c
1 -> ComplexAlgebraStr equals :: CLOPBAN2:def 7
ComplexAlgebraStr(#
(BoundedLinearOperators a1,a1),
(FuncMult a1),
(Add_ (BoundedLinearOperators a1,a1),(C_VectorSpace_of_LinearOperators a1,a1)),
(Mult_ (BoundedLinearOperators a1,a1),(C_VectorSpace_of_LinearOperators a1,a1)),
(FuncUnit a1),
(Zero_ (BoundedLinearOperators a1,a1),(C_VectorSpace_of_LinearOperators a1,a1)) #);
correctness
coherence
ComplexAlgebraStr(# (BoundedLinearOperators c1,c1),(FuncMult c1),(Add_ (BoundedLinearOperators c1,c1),(C_VectorSpace_of_LinearOperators c1,c1)),(Mult_ (BoundedLinearOperators c1,c1),(C_VectorSpace_of_LinearOperators c1,c1)),(FuncUnit c1),(Zero_ (BoundedLinearOperators c1,c1),(C_VectorSpace_of_LinearOperators c1,c1)) #) is ComplexAlgebraStr ;
;
end;
:: deftheorem Def7 defines C_Algebra_of_BoundedLinearOperators CLOPBAN2:def 7 :
for b
1 being
ComplexNormSpace holds
C_Algebra_of_BoundedLinearOperators b
1 = ComplexAlgebraStr(#
(BoundedLinearOperators b1,b1),
(FuncMult b1),
(Add_ (BoundedLinearOperators b1,b1),(C_VectorSpace_of_LinearOperators b1,b1)),
(Mult_ (BoundedLinearOperators b1,b1),(C_VectorSpace_of_LinearOperators b1,b1)),
(FuncUnit b1),
(Zero_ (BoundedLinearOperators b1,b1),(C_VectorSpace_of_LinearOperators b1,b1)) #);
theorem Th15: :: CLOPBAN2:15
theorem Th16: :: CLOPBAN2:16
theorem Th17: :: CLOPBAN2:17
theorem Th18: :: CLOPBAN2:18
definition
let c
1 be
ComplexNormSpace;
func C_Normed_Algebra_of_BoundedLinearOperators c
1 -> Normed_Complex_AlgebraStr equals :: CLOPBAN2:def 8
Normed_Complex_AlgebraStr(#
(BoundedLinearOperators a1,a1),
(FuncMult a1),
(Add_ (BoundedLinearOperators a1,a1),(C_VectorSpace_of_LinearOperators a1,a1)),
(Mult_ (BoundedLinearOperators a1,a1),(C_VectorSpace_of_LinearOperators a1,a1)),
(FuncUnit a1),
(Zero_ (BoundedLinearOperators a1,a1),(C_VectorSpace_of_LinearOperators a1,a1)),
(BoundedLinearOperatorsNorm a1,a1) #);
correctness
coherence
Normed_Complex_AlgebraStr(# (BoundedLinearOperators c1,c1),(FuncMult c1),(Add_ (BoundedLinearOperators c1,c1),(C_VectorSpace_of_LinearOperators c1,c1)),(Mult_ (BoundedLinearOperators c1,c1),(C_VectorSpace_of_LinearOperators c1,c1)),(FuncUnit c1),(Zero_ (BoundedLinearOperators c1,c1),(C_VectorSpace_of_LinearOperators c1,c1)),(BoundedLinearOperatorsNorm c1,c1) #) is Normed_Complex_AlgebraStr ;
;
end;
:: deftheorem Def8 defines C_Normed_Algebra_of_BoundedLinearOperators CLOPBAN2:def 8 :
for b
1 being
ComplexNormSpace holds
C_Normed_Algebra_of_BoundedLinearOperators b
1 = Normed_Complex_AlgebraStr(#
(BoundedLinearOperators b1,b1),
(FuncMult b1),
(Add_ (BoundedLinearOperators b1,b1),(C_VectorSpace_of_LinearOperators b1,b1)),
(Mult_ (BoundedLinearOperators b1,b1),(C_VectorSpace_of_LinearOperators b1,b1)),
(FuncUnit b1),
(Zero_ (BoundedLinearOperators b1,b1),(C_VectorSpace_of_LinearOperators b1,b1)),
(BoundedLinearOperatorsNorm b1,b1) #);
theorem Th19: :: CLOPBAN2:19
theorem Th20: :: CLOPBAN2:20
:: deftheorem Def9 defines Banach_Algebra-like_1 CLOPBAN2:def 9 :
:: deftheorem Def10 defines Banach_Algebra-like_2 CLOPBAN2:def 10 :
:: deftheorem Def11 defines Banach_Algebra-like_3 CLOPBAN2:def 11 :
:: deftheorem Def12 defines Banach_Algebra-like CLOPBAN2:def 12 :
theorem Th21: :: CLOPBAN2:21
theorem Th22: :: CLOPBAN2:22
theorem Th23: :: CLOPBAN2:23