Article CLOPBAN1, MML version 4.99.1005
:: CLOPBAN1:funcnot 1 => CLOPBAN1:func 1
definition
let a1 be set;
let a2 be non empty set;
let a3 be Function-like quasi_total Relation of [:COMPLEX,a2:],a2;
let a4 be complex set;
let a5 be Function-like quasi_total Relation of a1,a2;
redefine func a3 [;](a4,a5) -> Element of Funcs(a1,a2);
end;
:: CLOPBAN1:th 1
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct holds
ex b3 being Function-like quasi_total Relation of [:COMPLEX,Funcs(b1,the carrier of b2):],Funcs(b1,the carrier of b2) st
for b4 being Element of COMPLEX
for b5 being Element of Funcs(b1,the carrier of b2)
for b6 being Element of b1 holds
(b3 . [b4,b5]) . b6 = b4 * (b5 . b6);
:: CLOPBAN1:funcnot 2 => CLOPBAN1:func 2
definition
let a1 be non empty set;
let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
func FuncExtMult(A1,A2) -> Function-like quasi_total Relation of [:COMPLEX,Funcs(a1,the carrier of a2):],Funcs(a1,the carrier of a2) means
for b1 being Element of COMPLEX
for b2 being Element of Funcs(a1,the carrier of a2)
for b3 being Element of a1 holds
(it . [b1,b2]) . b3 = b1 * (b2 . b3);
end;
:: CLOPBAN1:def 1
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b3 being Function-like quasi_total Relation of [:COMPLEX,Funcs(b1,the carrier of b2):],Funcs(b1,the carrier of b2) holds
b3 = FuncExtMult(b1,b2)
iff
for b4 being Element of COMPLEX
for b5 being Element of Funcs(b1,the carrier of b2)
for b6 being Element of b1 holds
(b3 . [b4,b5]) . b6 = b4 * (b5 . b6);
:: CLOPBAN1:th 2
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b3 being Element of b1 holds
(FuncZero(b1,b2)) . b3 = 0. b2;
:: CLOPBAN1:th 3
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b3, b4 being Element of Funcs(b1,the carrier of b2)
for b5 being Element of COMPLEX holds
b3 = (FuncExtMult(b1,b2)) . [b5,b4]
iff
for b6 being Element of b1 holds
b3 . b6 = b5 * (b4 . b6);
:: CLOPBAN1:th 4
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b3, b4 being Element of Funcs(b1,the carrier of b2) holds
(FuncAdd(b1,b2)) .(b3,b4) = (FuncAdd(b1,b2)) .(b4,b3);
:: CLOPBAN1:th 5
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b3, b4, b5 being Element of Funcs(b1,the carrier of b2) holds
(FuncAdd(b1,b2)) .(b3,(FuncAdd(b1,b2)) .(b4,b5)) = (FuncAdd(b1,b2)) .((FuncAdd(b1,b2)) .(b3,b4),b5);
:: CLOPBAN1:th 6
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b3 being Element of Funcs(b1,the carrier of b2) holds
(FuncAdd(b1,b2)) .(FuncZero(b1,b2),b3) = b3;
:: CLOPBAN1:th 7
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b3 being Element of Funcs(b1,the carrier of b2) holds
(FuncAdd(b1,b2)) .(b3,(FuncExtMult(b1,b2)) . [- 1r,b3]) = FuncZero(b1,b2);
:: CLOPBAN1:th 8
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b3 being Element of Funcs(b1,the carrier of b2) holds
(FuncExtMult(b1,b2)) . [1r,b3] = b3;
:: CLOPBAN1:th 9
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b3 being Element of Funcs(b1,the carrier of b2)
for b4, b5 being Element of COMPLEX holds
(FuncExtMult(b1,b2)) . [b4,(FuncExtMult(b1,b2)) . [b5,b3]] = (FuncExtMult(b1,b2)) . [b4 * b5,b3];
:: CLOPBAN1:th 10
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b3 being Element of Funcs(b1,the carrier of b2)
for b4, b5 being Element of COMPLEX holds
(FuncAdd(b1,b2)) .((FuncExtMult(b1,b2)) . [b4,b3],(FuncExtMult(b1,b2)) . [b5,b3]) = (FuncExtMult(b1,b2)) . [b4 + b5,b3];
:: CLOPBAN1:th 11
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct holds
CLSStruct(#Funcs(b1,the carrier of b2),FuncZero(b1,b2),FuncAdd(b1,b2),FuncExtMult(b1,b2)#) is non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
:: CLOPBAN1:funcnot 3 => CLOPBAN1:func 3
definition
let a1 be non empty set;
let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
func ComplexVectSpace(A1,A2) -> non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct equals
CLSStruct(#Funcs(a1,the carrier of a2),FuncZero(a1,a2),FuncAdd(a1,a2),FuncExtMult(a1,a2)#);
end;
:: CLOPBAN1:def 2
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct holds
ComplexVectSpace(b1,b2) = CLSStruct(#Funcs(b1,the carrier of b2),FuncZero(b1,b2),FuncAdd(b1,b2),FuncExtMult(b1,b2)#);
:: CLOPBAN1:funcreg 1
registration
let a1 be non empty set;
let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
cluster ComplexVectSpace(a1,a2) -> non empty right_complementable Abelian add-associative right_zeroed strict ComplexLinearSpace-like;
end;
:: CLOPBAN1:funcreg 2
registration
let a1 be non empty set;
let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
cluster ComplexVectSpace(a1,a2) -> non empty right_complementable Abelian add-associative right_zeroed constituted-Functions ComplexLinearSpace-like;
end;
:: CLOPBAN1:funcnot 4 => CLOPBAN1:func 4
definition
let a1 be non empty set;
let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
let a3 be Element of the carrier of ComplexVectSpace(a1,a2);
let a4 be Element of a1;
redefine func a3 . a4 -> Element of the carrier of a2;
end;
:: CLOPBAN1:th 12
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b3, b4, b5 being Element of the carrier of ComplexVectSpace(b1,b2) holds
b5 = b3 + b4
iff
for b6 being Element of b1 holds
b5 . b6 = (b3 . b6) + (b4 . b6);
:: CLOPBAN1:th 13
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b3, b4 being Element of the carrier of ComplexVectSpace(b1,b2)
for b5 being Element of COMPLEX holds
b4 = b5 * b3
iff
for b6 being Element of b1 holds
b4 . b6 = b5 * (b3 . b6);
:: CLOPBAN1:attrnot 1 => CLOPBAN1:attr 1
definition
let a1 be non empty CLSStruct;
let a2 be non empty addLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
attr a3 is additive means
for b1, b2 being Element of the carrier of a1 holds
a3 . (b1 + b2) = (a3 . b1) + (a3 . b2);
end;
:: CLOPBAN1:dfs 3
definiens
let a1 be non empty CLSStruct;
let a2 be non empty addLoopStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
a3 is additive
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1 holds
a3 . (b1 + b2) = (a3 . b1) + (a3 . b2);
:: CLOPBAN1:def 3
theorem
for b1 being non empty CLSStruct
for b2 being non empty addLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is additive(b1, b2)
iff
for b4, b5 being Element of the carrier of b1 holds
b3 . (b4 + b5) = (b3 . b4) + (b3 . b5);
:: CLOPBAN1:attrnot 2 => CLOPBAN1:attr 2
definition
let a1, a2 be non empty CLSStruct;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
attr a3 is homogeneous means
for b1 being Element of the carrier of a1
for b2 being Element of COMPLEX holds
a3 . (b2 * b1) = b2 * (a3 . b1);
end;
:: CLOPBAN1:dfs 4
definiens
let a1, a2 be non empty CLSStruct;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
a3 is homogeneous
it is sufficient to prove
thus for b1 being Element of the carrier of a1
for b2 being Element of COMPLEX holds
a3 . (b2 * b1) = b2 * (a3 . b1);
:: CLOPBAN1:def 4
theorem
for b1, b2 being non empty CLSStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is homogeneous(b1, b2)
iff
for b4 being Element of the carrier of b1
for b5 being Element of COMPLEX holds
b3 . (b5 * b4) = b5 * (b3 . b4);
:: CLOPBAN1:exreg 1
registration
let a1 be non empty CLSStruct;
let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
cluster non empty Relation-like Function-like quasi_total total additive homogeneous Relation of the carrier of a1,the carrier of a2;
end;
:: CLOPBAN1:modenot 1
definition
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
mode LinearOperator of a1,a2 is Function-like quasi_total additive homogeneous Relation of the carrier of a1,the carrier of a2;
end;
:: CLOPBAN1:funcnot 5 => CLOPBAN1:func 5
definition
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
func LinearOperators(A1,A2) -> Element of bool the carrier of ComplexVectSpace(the carrier of a1,a2) means
for b1 being set holds
b1 in it
iff
b1 is Function-like quasi_total additive homogeneous Relation of the carrier of a1,the carrier of a2;
end;
:: CLOPBAN1:def 5
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b3 being Element of bool the carrier of ComplexVectSpace(the carrier of b1,b2) holds
b3 = LinearOperators(b1,b2)
iff
for b4 being set holds
b4 in b3
iff
b4 is Function-like quasi_total additive homogeneous Relation of the carrier of b1,the carrier of b2;
:: CLOPBAN1:funcreg 3
registration
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
cluster LinearOperators(a1,a2) -> non empty functional;
end;
:: CLOPBAN1:th 15
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct holds
LinearOperators(b1,b2) is linearly-closed(ComplexVectSpace(the carrier of b1,b2));
:: CLOPBAN1:th 16
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct 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 b1,b2);
:: CLOPBAN1:funcreg 4
registration
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
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))#) -> right_complementable Abelian add-associative right_zeroed strict ComplexLinearSpace-like;
end;
:: CLOPBAN1:funcnot 6 => CLOPBAN1:func 6
definition
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
func C_VectorSpace_of_LinearOperators(A1,A2) -> non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct equals
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))#);
end;
:: CLOPBAN1:def 6
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct holds
C_VectorSpace_of_LinearOperators(b1,b2) = 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))#);
:: CLOPBAN1:funcreg 5
registration
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
cluster C_VectorSpace_of_LinearOperators(a1,a2) -> non empty right_complementable Abelian add-associative right_zeroed strict ComplexLinearSpace-like;
end;
:: CLOPBAN1:funcreg 6
registration
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
cluster C_VectorSpace_of_LinearOperators(a1,a2) -> non empty right_complementable Abelian add-associative right_zeroed constituted-Functions ComplexLinearSpace-like;
end;
:: CLOPBAN1:funcnot 7 => CLOPBAN1:func 7
definition
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
let a3 be Element of the carrier of C_VectorSpace_of_LinearOperators(a1,a2);
let a4 be Element of the carrier of a1;
redefine func a3 . a4 -> Element of the carrier of a2;
end;
:: CLOPBAN1:th 18
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b3, b4, b5 being Element of the carrier of C_VectorSpace_of_LinearOperators(b1,b2) holds
b5 = b3 + b4
iff
for b6 being Element of the carrier of b1 holds
b5 . b6 = (b3 . b6) + (b4 . b6);
:: CLOPBAN1:th 19
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b3, b4 being Element of the carrier of C_VectorSpace_of_LinearOperators(b1,b2)
for b5 being Element of COMPLEX holds
b4 = b5 * b3
iff
for b6 being Element of the carrier of b1 holds
b4 . b6 = b5 * (b3 . b6);
:: CLOPBAN1:th 20
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct holds
0. C_VectorSpace_of_LinearOperators(b1,b2) = (the carrier of b1) --> 0. b2;
:: CLOPBAN1:th 21
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct holds
(the carrier of b1) --> 0. b2 is Function-like quasi_total additive homogeneous Relation of the carrier of b1,the carrier of b2;
:: CLOPBAN1:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1
st b2 is convergent(b1) & lim b2 = b3
holds ||.b2.|| is convergent & lim ||.b2.|| = ||.b3.||;
:: CLOPBAN1:attrnot 3 => CLOPBAN1:attr 3
definition
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a3 be Function-like quasi_total additive homogeneous Relation of the carrier of a1,the carrier of a2;
attr a3 is bounded means
ex b1 being Element of REAL st
0 <= b1 &
(for b2 being Element of the carrier of a1 holds
||.a3 . b2.|| <= b1 * ||.b2.||);
end;
:: CLOPBAN1:dfs 7
definiens
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a3 be Function-like quasi_total additive homogeneous Relation of the carrier of a1,the carrier of a2;
To prove
a3 is bounded
it is sufficient to prove
thus ex b1 being Element of REAL st
0 <= b1 &
(for b2 being Element of the carrier of a1 holds
||.a3 . b2.|| <= b1 * ||.b2.||);
:: CLOPBAN1:def 7
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like quasi_total additive homogeneous Relation of the carrier of b1,the carrier of b2 holds
b3 is bounded(b1, b2)
iff
ex b4 being Element of REAL st
0 <= b4 &
(for b5 being Element of the carrier of b1 holds
||.b3 . b5.|| <= b4 * ||.b5.||);
:: CLOPBAN1:th 23
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like quasi_total additive homogeneous Relation of the carrier of b1,the carrier of b2
st for b4 being Element of the carrier of b1 holds
b3 . b4 = 0. b2
holds b3 is bounded(b1, b2);
:: CLOPBAN1:exreg 2
registration
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
cluster non empty Relation-like Function-like quasi_total total additive homogeneous bounded Relation of the carrier of a1,the carrier of a2;
end;
:: CLOPBAN1:funcnot 8 => CLOPBAN1:func 8
definition
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
func BoundedLinearOperators(A1,A2) -> Element of bool the carrier of C_VectorSpace_of_LinearOperators(a1,a2) means
for b1 being set holds
b1 in it
iff
b1 is Function-like quasi_total additive homogeneous bounded Relation of the carrier of a1,the carrier of a2;
end;
:: CLOPBAN1:def 8
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Element of bool the carrier of C_VectorSpace_of_LinearOperators(b1,b2) holds
b3 = BoundedLinearOperators(b1,b2)
iff
for b4 being set holds
b4 in b3
iff
b4 is Function-like quasi_total additive homogeneous bounded Relation of the carrier of b1,the carrier of b2;
:: CLOPBAN1:funcreg 7
registration
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
cluster BoundedLinearOperators(a1,a2) -> non empty;
end;
:: CLOPBAN1:th 24
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR holds
BoundedLinearOperators(b1,b2) is linearly-closed(C_VectorSpace_of_LinearOperators(b1,b2));
:: CLOPBAN1:th 25
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR 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(b1,b2);
:: CLOPBAN1:funcreg 8
registration
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
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))#) -> right_complementable Abelian add-associative right_zeroed strict ComplexLinearSpace-like;
end;
:: CLOPBAN1:funcnot 9 => CLOPBAN1:func 9
definition
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
func C_VectorSpace_of_BoundedLinearOperators(A1,A2) -> non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct equals
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))#);
end;
:: CLOPBAN1:def 9
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR holds
C_VectorSpace_of_BoundedLinearOperators(b1,b2) = 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))#);
:: CLOPBAN1:funcreg 9
registration
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
cluster C_VectorSpace_of_BoundedLinearOperators(a1,a2) -> non empty right_complementable Abelian add-associative right_zeroed strict ComplexLinearSpace-like;
end;
:: CLOPBAN1:condreg 1
registration
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
cluster -> Relation-like Function-like (Element of the carrier of C_VectorSpace_of_BoundedLinearOperators(a1,a2));
end;
:: CLOPBAN1:funcnot 10 => CLOPBAN1:func 10
definition
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a3 be Element of the carrier of C_VectorSpace_of_BoundedLinearOperators(a1,a2);
let a4 be Element of the carrier of a1;
redefine func a3 . a4 -> Element of the carrier of a2;
end;
:: CLOPBAN1:th 27
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3, b4, b5 being Element of the carrier of C_VectorSpace_of_BoundedLinearOperators(b1,b2) holds
b5 = b3 + b4
iff
for b6 being Element of the carrier of b1 holds
b5 . b6 = (b3 . b6) + (b4 . b6);
:: CLOPBAN1:th 28
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3, b4 being Element of the carrier of C_VectorSpace_of_BoundedLinearOperators(b1,b2)
for b5 being Element of COMPLEX holds
b4 = b5 * b3
iff
for b6 being Element of the carrier of b1 holds
b4 . b6 = b5 * (b3 . b6);
:: CLOPBAN1:th 29
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR holds
0. C_VectorSpace_of_BoundedLinearOperators(b1,b2) = (the carrier of b1) --> 0. b2;
:: CLOPBAN1:funcnot 11 => CLOPBAN1:func 11
definition
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a3 be set;
assume a3 in BoundedLinearOperators(a1,a2);
func modetrans(A3,A1,A2) -> Function-like quasi_total additive homogeneous bounded Relation of the carrier of a1,the carrier of a2 equals
a3;
end;
:: CLOPBAN1:def 10
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being set
st b3 in BoundedLinearOperators(b1,b2)
holds modetrans(b3,b1,b2) = b3;
:: CLOPBAN1:funcnot 12 => CLOPBAN1:func 12
definition
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a3 be Function-like quasi_total additive homogeneous Relation of the carrier of a1,the carrier of a2;
func PreNorms A3 -> non empty Element of bool REAL equals
{||.a3 . b1.|| where b1 is Element of the carrier of a1: ||.b1.|| <= 1};
end;
:: CLOPBAN1:def 11
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like quasi_total additive homogeneous Relation of the carrier of b1,the carrier of b2 holds
PreNorms b3 = {||.b3 . b4.|| where b4 is Element of the carrier of b1: ||.b4.|| <= 1};
:: CLOPBAN1:th 30
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like quasi_total additive homogeneous bounded Relation of the carrier of b1,the carrier of b2 holds
PreNorms b3 is not empty & PreNorms b3 is bounded_above;
:: CLOPBAN1:th 31
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like quasi_total additive homogeneous Relation of the carrier of b1,the carrier of b2 holds
b3 is bounded(b1, b2)
iff
PreNorms b3 is bounded_above;
:: CLOPBAN1:th 32
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR holds
ex b3 being Function-like quasi_total Relation of BoundedLinearOperators(b1,b2),REAL st
for b4 being set
st b4 in BoundedLinearOperators(b1,b2)
holds b3 . b4 = sup PreNorms modetrans(b4,b1,b2);
:: CLOPBAN1:funcnot 13 => CLOPBAN1:func 13
definition
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
func BoundedLinearOperatorsNorm(A1,A2) -> Function-like quasi_total Relation of BoundedLinearOperators(a1,a2),REAL means
for b1 being set
st b1 in BoundedLinearOperators(a1,a2)
holds it . b1 = sup PreNorms modetrans(b1,a1,a2);
end;
:: CLOPBAN1:def 12
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like quasi_total Relation of BoundedLinearOperators(b1,b2),REAL holds
b3 = BoundedLinearOperatorsNorm(b1,b2)
iff
for b4 being set
st b4 in BoundedLinearOperators(b1,b2)
holds b3 . b4 = sup PreNorms modetrans(b4,b1,b2);
:: CLOPBAN1:th 33
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like quasi_total additive homogeneous bounded Relation of the carrier of b1,the carrier of b2 holds
modetrans(b3,b1,b2) = b3;
:: CLOPBAN1:th 34
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like quasi_total additive homogeneous bounded Relation of the carrier of b1,the carrier of b2 holds
(BoundedLinearOperatorsNorm(b1,b2)) . b3 = sup PreNorms b3;
:: CLOPBAN1:funcnot 14 => CLOPBAN1:func 14
definition
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
func C_NormSpace_of_BoundedLinearOperators(A1,A2) -> non empty CNORMSTR equals
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)#);
end;
:: CLOPBAN1:def 13
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR holds
C_NormSpace_of_BoundedLinearOperators(b1,b2) = 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)#);
:: CLOPBAN1:th 35
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR holds
(the carrier of b1) --> 0. b2 = 0. C_NormSpace_of_BoundedLinearOperators(b1,b2);
:: CLOPBAN1:th 36
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Element of the carrier of C_NormSpace_of_BoundedLinearOperators(b1,b2)
for b4 being Function-like quasi_total additive homogeneous bounded Relation of the carrier of b1,the carrier of b2
st b4 = b3
for b5 being Element of the carrier of b1 holds
||.b4 . b5.|| <= ||.b3.|| * ||.b5.||;
:: CLOPBAN1:th 37
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Element of the carrier of C_NormSpace_of_BoundedLinearOperators(b1,b2) holds
0 <= ||.b3.||;
:: CLOPBAN1:th 38
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Element of the carrier of C_NormSpace_of_BoundedLinearOperators(b1,b2)
st b3 = 0. C_NormSpace_of_BoundedLinearOperators(b1,b2)
holds 0 = ||.b3.||;
:: CLOPBAN1:condreg 2
registration
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
cluster -> Relation-like Function-like (Element of the carrier of C_NormSpace_of_BoundedLinearOperators(a1,a2));
end;
:: CLOPBAN1:funcnot 15 => CLOPBAN1:func 15
definition
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a3 be Element of the carrier of C_NormSpace_of_BoundedLinearOperators(a1,a2);
let a4 be Element of the carrier of a1;
redefine func a3 . a4 -> Element of the carrier of a2;
end;
:: CLOPBAN1:th 39
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3, b4, b5 being Element of the carrier of C_NormSpace_of_BoundedLinearOperators(b1,b2) holds
b5 = b3 + b4
iff
for b6 being Element of the carrier of b1 holds
b5 . b6 = (b3 . b6) + (b4 . b6);
:: CLOPBAN1:th 40
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3, b4 being Element of the carrier of C_NormSpace_of_BoundedLinearOperators(b1,b2)
for b5 being Element of COMPLEX holds
b4 = b5 * b3
iff
for b6 being Element of the carrier of b1 holds
b4 . b6 = b5 * (b3 . b6);
:: CLOPBAN1:th 41
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3, b4 being Element of the carrier of C_NormSpace_of_BoundedLinearOperators(b1,b2)
for b5 being Element of COMPLEX holds
(||.b3.|| = 0 implies b3 = 0. C_NormSpace_of_BoundedLinearOperators(b1,b2)) &
(b3 = 0. C_NormSpace_of_BoundedLinearOperators(b1,b2) implies ||.b3.|| = 0) &
||.b5 * b3.|| = |.b5.| * ||.b3.|| &
||.b3 + b4.|| <= ||.b3.|| + ||.b4.||;
:: CLOPBAN1:th 42
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR holds
C_NormSpace_of_BoundedLinearOperators(b1,b2) is ComplexNormSpace-like;
:: CLOPBAN1:th 43
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR holds
C_NormSpace_of_BoundedLinearOperators(b1,b2) is non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
:: CLOPBAN1:funcreg 10
registration
let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
cluster C_NormSpace_of_BoundedLinearOperators(a1,a2) -> non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like;
end;
:: CLOPBAN1:th 44
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3, b4, b5 being Element of the carrier of C_NormSpace_of_BoundedLinearOperators(b1,b2) holds
b5 = b3 - b4
iff
for b6 being Element of the carrier of b1 holds
b5 . b6 = (b3 . b6) - (b4 . b6);
:: CLOPBAN1:attrnot 4 => CLOPBAN1:attr 4
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
attr a1 is complete means
for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
st b1 is CCauchy(a1)
holds b1 is convergent(a1);
end;
:: CLOPBAN1:dfs 14
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
To prove
a1 is complete
it is sufficient to prove
thus for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
st b1 is CCauchy(a1)
holds b1 is convergent(a1);
:: CLOPBAN1:def 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR holds
b1 is complete
iff
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is CCauchy(b1)
holds b2 is convergent(b1);
:: CLOPBAN1:exreg 3
registration
cluster non empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed ComplexLinearSpace-like ComplexNormSpace-like complete CNORMSTR;
end;
:: CLOPBAN1:modenot 2
definition
mode ComplexBanachSpace is non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like complete CNORMSTR;
end;
:: CLOPBAN1:th 45
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1)
holds ||.b2.|| is convergent &
lim ||.b2.|| = ||.lim b2.||;
:: CLOPBAN1:th 46
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
st b2 is complete
for b3 being Function-like quasi_total Relation of NAT,the carrier of C_NormSpace_of_BoundedLinearOperators(b1,b2)
st b3 is CCauchy(C_NormSpace_of_BoundedLinearOperators(b1,b2))
holds b3 is convergent(C_NormSpace_of_BoundedLinearOperators(b1,b2));
:: CLOPBAN1:th 47
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like complete CNORMSTR holds
C_NormSpace_of_BoundedLinearOperators(b1,b2) is non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like complete CNORMSTR;
:: CLOPBAN1:funcreg 11
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like complete CNORMSTR;
cluster C_NormSpace_of_BoundedLinearOperators(a1,a2) -> non empty complete;
end;