Article LOPBAN_2, MML version 4.99.1005
:: LOPBAN_2:th 1
theorem
for b1, b2, b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b4 being Function-like quasi_total additive homogeneous Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total additive homogeneous Relation of the carrier of b2,the carrier of b3 holds
b5 * b4 is Function-like quasi_total additive homogeneous Relation of the carrier of b1,the carrier of b3;
:: LOPBAN_2:th 2
theorem
for b1, b2, b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like quasi_total additive homogeneous bounded Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total additive homogeneous bounded Relation of the carrier of b2,the carrier of b3 holds
b5 * b4 is Function-like quasi_total additive homogeneous bounded Relation of the carrier of b1,the carrier of b3 &
(for b6 being Element of the carrier of b1 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));
:: LOPBAN_2:funcnot 1 => LOPBAN_2:func 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a2, a3 be Function-like quasi_total additive homogeneous bounded Relation of the carrier of a1,the carrier of a1;
redefine func a3 * a2 -> Function-like quasi_total additive homogeneous bounded Relation of the carrier of a1,the carrier of a1;
end;
:: LOPBAN_2:funcnot 2 => LOPBAN_2:func 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a2, a3 be Element of BoundedLinearOperators(a1,a1);
func A2 + A3 -> Element of BoundedLinearOperators(a1,a1) equals
(Add_(BoundedLinearOperators(a1,a1),R_VectorSpace_of_LinearOperators(a1,a1))) .(a2,a3);
end;
:: LOPBAN_2:def 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Element of BoundedLinearOperators(b1,b1) holds
b2 + b3 = (Add_(BoundedLinearOperators(b1,b1),R_VectorSpace_of_LinearOperators(b1,b1))) .(b2,b3);
:: LOPBAN_2:funcnot 3 => LOPBAN_2:func 3
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a2, a3 be Element of BoundedLinearOperators(a1,a1);
func A3 * A2 -> Element of BoundedLinearOperators(a1,a1) equals
(modetrans(a3,a1,a1)) * modetrans(a2,a1,a1);
end;
:: LOPBAN_2:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Element of BoundedLinearOperators(b1,b1) holds
b3 * b2 = (modetrans(b3,b1,b1)) * modetrans(b2,b1,b1);
:: LOPBAN_2:funcnot 4 => LOPBAN_2:func 4
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
let a2 be Element of BoundedLinearOperators(a1,a1);
let a3 be Element of REAL;
func A3 * A2 -> Element of BoundedLinearOperators(a1,a1) equals
(Mult_(BoundedLinearOperators(a1,a1),R_VectorSpace_of_LinearOperators(a1,a1))) .(a3,a2);
end;
:: LOPBAN_2:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of BoundedLinearOperators(b1,b1)
for b3 being Element of REAL holds
b3 * b2 = (Mult_(BoundedLinearOperators(b1,b1),R_VectorSpace_of_LinearOperators(b1,b1))) .(b3,b2);
:: LOPBAN_2:funcnot 5 => LOPBAN_2:func 5
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
func FuncMult A1 -> Function-like quasi_total Relation of [:BoundedLinearOperators(a1,a1),BoundedLinearOperators(a1,a1):],BoundedLinearOperators(a1,a1) means
for b1, b2 being Element of BoundedLinearOperators(a1,a1) holds
it .(b1,b2) = b1 * b2;
end;
:: LOPBAN_2:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of [:BoundedLinearOperators(b1,b1),BoundedLinearOperators(b1,b1):],BoundedLinearOperators(b1,b1) holds
b2 = FuncMult b1
iff
for b3, b4 being Element of BoundedLinearOperators(b1,b1) holds
b2 .(b3,b4) = b3 * b4;
:: LOPBAN_2:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
id the carrier of b1 is Function-like quasi_total additive homogeneous bounded Relation of the carrier of b1,the carrier of b1;
:: LOPBAN_2:funcnot 6 => LOPBAN_2:func 6
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
func FuncUnit A1 -> Element of BoundedLinearOperators(a1,a1) equals
id the carrier of a1;
end;
:: LOPBAN_2:def 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
FuncUnit b1 = id the carrier of b1;
:: LOPBAN_2:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3, b4 being Function-like quasi_total additive homogeneous bounded Relation of the carrier of b1,the carrier of b1 holds
b4 = b2 * b3
iff
for b5 being Element of the carrier of b1 holds
b4 . b5 = b2 . (b3 . b5);
:: LOPBAN_2:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3, b4 being Function-like quasi_total additive homogeneous bounded Relation of the carrier of b1,the carrier of b1 holds
b2 * (b3 * b4) = (b2 * b3) * b4;
:: LOPBAN_2:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total additive homogeneous bounded Relation of the carrier of b1,the carrier of b1 holds
b2 * id the carrier of b1 = b2 & (id the carrier of b1) * b2 = b2;
:: LOPBAN_2:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3, b4 being Element of BoundedLinearOperators(b1,b1) holds
b2 * (b3 * b4) = (b2 * b3) * b4;
:: LOPBAN_2:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of BoundedLinearOperators(b1,b1) holds
b2 * FuncUnit b1 = b2 & (FuncUnit b1) * b2 = b2;
:: LOPBAN_2:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3, b4 being Element of BoundedLinearOperators(b1,b1) holds
b2 * (b3 + b4) = (b2 * b3) + (b2 * b4);
:: LOPBAN_2:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3, b4 being Element of BoundedLinearOperators(b1,b1) holds
(b3 + b4) * b2 = (b3 * b2) + (b4 * b2);
:: LOPBAN_2:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Element of BoundedLinearOperators(b1,b1)
for b4, b5 being Element of REAL holds
(b4 * b5) * (b2 * b3) = (b4 * b2) * (b5 * b3);
:: LOPBAN_2:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Element of BoundedLinearOperators(b1,b1)
for b4 being Element of REAL holds
b4 * (b2 * b3) = (b4 * b2) * b3;
:: LOPBAN_2:funcnot 7 => LOPBAN_2:func 7
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
func Ring_of_BoundedLinearOperators A1 -> doubleLoopStr equals
doubleLoopStr(#BoundedLinearOperators(a1,a1),Add_(BoundedLinearOperators(a1,a1),R_VectorSpace_of_LinearOperators(a1,a1)),FuncMult a1,FuncUnit a1,Zero_(BoundedLinearOperators(a1,a1),R_VectorSpace_of_LinearOperators(a1,a1))#);
end;
:: LOPBAN_2:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
Ring_of_BoundedLinearOperators b1 = doubleLoopStr(#BoundedLinearOperators(b1,b1),Add_(BoundedLinearOperators(b1,b1),R_VectorSpace_of_LinearOperators(b1,b1)),FuncMult b1,FuncUnit b1,Zero_(BoundedLinearOperators(b1,b1),R_VectorSpace_of_LinearOperators(b1,b1))#);
:: LOPBAN_2:funcreg 1
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
cluster Ring_of_BoundedLinearOperators a1 -> non empty strict;
end;
:: LOPBAN_2:funcreg 2
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
cluster Ring_of_BoundedLinearOperators a1 -> unital;
end;
:: LOPBAN_2:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3, b4 being Element of the carrier of Ring_of_BoundedLinearOperators b1 holds
b2 + b3 = b3 + b2 &
(b2 + b3) + b4 = b2 + (b3 + b4) &
b2 + 0. Ring_of_BoundedLinearOperators b1 = b2 &
(ex b5 being Element of the carrier of Ring_of_BoundedLinearOperators b1 st
b2 + b5 = 0. Ring_of_BoundedLinearOperators b1) &
(b2 * b3) * b4 = b2 * (b3 * b4) &
b2 * 1. Ring_of_BoundedLinearOperators b1 = b2 &
(1. Ring_of_BoundedLinearOperators b1) * b2 = b2 &
b2 * (b3 + b4) = (b2 * b3) + (b2 * b4) &
(b3 + b4) * b2 = (b3 * b2) + (b4 * b2);
:: LOPBAN_2:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
Ring_of_BoundedLinearOperators b1 is non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
:: LOPBAN_2:funcreg 3
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
cluster Ring_of_BoundedLinearOperators a1 -> right_complementable Abelian add-associative right_zeroed associative right_unital distributive left_unital;
end;
:: LOPBAN_2:funcnot 8 => LOPBAN_2:func 8
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
func R_Algebra_of_BoundedLinearOperators A1 -> AlgebraStr equals
AlgebraStr(#BoundedLinearOperators(a1,a1),FuncMult a1,Add_(BoundedLinearOperators(a1,a1),R_VectorSpace_of_LinearOperators(a1,a1)),Mult_(BoundedLinearOperators(a1,a1),R_VectorSpace_of_LinearOperators(a1,a1)),FuncUnit a1,Zero_(BoundedLinearOperators(a1,a1),R_VectorSpace_of_LinearOperators(a1,a1))#);
end;
:: LOPBAN_2:def 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
R_Algebra_of_BoundedLinearOperators b1 = AlgebraStr(#BoundedLinearOperators(b1,b1),FuncMult b1,Add_(BoundedLinearOperators(b1,b1),R_VectorSpace_of_LinearOperators(b1,b1)),Mult_(BoundedLinearOperators(b1,b1),R_VectorSpace_of_LinearOperators(b1,b1)),FuncUnit b1,Zero_(BoundedLinearOperators(b1,b1),R_VectorSpace_of_LinearOperators(b1,b1))#);
:: LOPBAN_2:funcreg 4
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
cluster R_Algebra_of_BoundedLinearOperators a1 -> non empty strict;
end;
:: LOPBAN_2:funcreg 5
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
cluster R_Algebra_of_BoundedLinearOperators a1 -> unital;
end;
:: LOPBAN_2:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3, b4 being Element of the carrier of R_Algebra_of_BoundedLinearOperators b1
for b5, b6 being Element of REAL holds
b2 + b3 = b3 + b2 &
(b2 + b3) + b4 = b2 + (b3 + b4) &
b2 + 0. R_Algebra_of_BoundedLinearOperators b1 = b2 &
(ex b7 being Element of the carrier of R_Algebra_of_BoundedLinearOperators b1 st
b2 + b7 = 0. R_Algebra_of_BoundedLinearOperators b1) &
(b2 * b3) * b4 = b2 * (b3 * b4) &
b2 * 1. R_Algebra_of_BoundedLinearOperators b1 = b2 &
(1. R_Algebra_of_BoundedLinearOperators b1) * b2 = b2 &
b2 * (b3 + b4) = (b2 * b3) + (b2 * b4) &
(b3 + b4) * b2 = (b3 * b2) + (b4 * b2) &
b5 * (b2 * b3) = (b5 * b2) * b3 &
b5 * (b2 + b3) = (b5 * b2) + (b5 * b3) &
(b5 + b6) * b2 = (b5 * b2) + (b6 * b2) &
(b5 * b6) * b2 = b5 * (b6 * b2) &
(b5 * b6) * (b2 * b3) = (b5 * b2) * (b6 * b3);
:: LOPBAN_2:modenot 1
definition
mode BLAlgebra is non empty right_complementable Abelian add-associative right_zeroed Algebra-like associative right-distributive right_unital AlgebraStr;
end;
:: LOPBAN_2:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
R_Algebra_of_BoundedLinearOperators b1 is non empty right_complementable Abelian add-associative right_zeroed Algebra-like associative right-distributive right_unital AlgebraStr;
:: LOPBAN_2:funcreg 6
registration
cluster l1_Space -> non empty complete;
end;
:: LOPBAN_2:funcreg 7
registration
cluster l1_Space -> non empty non trivial;
end;
:: LOPBAN_2:exreg 1
registration
cluster non empty non trivial left_complementable right_complementable complementable Abelian add-associative right_zeroed RealLinearSpace-like zeroed RealNormSpace-like complete NORMSTR;
end;
:: LOPBAN_2:th 17
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
ex b2 being Element of the carrier of b1 st
||.b2.|| = 1;
:: LOPBAN_2:th 18
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
(BoundedLinearOperatorsNorm(b1,b1)) . id the carrier of b1 = 1;
:: LOPBAN_2:structnot 1 => LOPBAN_2:struct 1
definition
struct(AlgebraStr, NORMSTR) Normed_AlgebraStr(#
carrier -> set,
multF -> Function-like quasi_total Relation of [:the carrier of it,the carrier of it:],the carrier of it,
addF -> Function-like quasi_total Relation of [:the carrier of it,the carrier of it:],the carrier of it,
Mult -> Function-like quasi_total Relation of [:REAL,the carrier of it:],the carrier of it,
OneF -> Element of the carrier of it,
ZeroF -> Element of the carrier of it,
norm -> Function-like quasi_total Relation of the carrier of it,REAL
#);
end;
:: LOPBAN_2:attrnot 1 => LOPBAN_2:attr 1
definition
let a1 be Normed_AlgebraStr;
attr a1 is strict;
end;
:: LOPBAN_2:exreg 2
registration
cluster strict Normed_AlgebraStr;
end;
:: LOPBAN_2:aggrnot 1 => LOPBAN_2:aggr 1
definition
let a1 be set;
let a2, a3 be Function-like quasi_total Relation of [:a1,a1:],a1;
let a4 be Function-like quasi_total Relation of [:REAL,a1:],a1;
let a5, a6 be Element of a1;
let a7 be Function-like quasi_total Relation of a1,REAL;
aggr Normed_AlgebraStr(#a1,a2,a3,a4,a5,a6,a7#) -> strict Normed_AlgebraStr;
end;
:: LOPBAN_2:exreg 3
registration
cluster non empty Normed_AlgebraStr;
end;
:: LOPBAN_2:funcnot 9 => LOPBAN_2:func 9
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
func R_Normed_Algebra_of_BoundedLinearOperators A1 -> Normed_AlgebraStr equals
Normed_AlgebraStr(#BoundedLinearOperators(a1,a1),FuncMult a1,Add_(BoundedLinearOperators(a1,a1),R_VectorSpace_of_LinearOperators(a1,a1)),Mult_(BoundedLinearOperators(a1,a1),R_VectorSpace_of_LinearOperators(a1,a1)),FuncUnit a1,Zero_(BoundedLinearOperators(a1,a1),R_VectorSpace_of_LinearOperators(a1,a1)),BoundedLinearOperatorsNorm(a1,a1)#);
end;
:: LOPBAN_2:def 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
R_Normed_Algebra_of_BoundedLinearOperators b1 = Normed_AlgebraStr(#BoundedLinearOperators(b1,b1),FuncMult b1,Add_(BoundedLinearOperators(b1,b1),R_VectorSpace_of_LinearOperators(b1,b1)),Mult_(BoundedLinearOperators(b1,b1),R_VectorSpace_of_LinearOperators(b1,b1)),FuncUnit b1,Zero_(BoundedLinearOperators(b1,b1),R_VectorSpace_of_LinearOperators(b1,b1)),BoundedLinearOperatorsNorm(b1,b1)#);
:: LOPBAN_2:funcreg 8
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
cluster R_Normed_Algebra_of_BoundedLinearOperators a1 -> non empty strict;
end;
:: LOPBAN_2:funcreg 9
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
cluster R_Normed_Algebra_of_BoundedLinearOperators a1 -> unital;
end;
:: LOPBAN_2:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3, b4 being Element of the carrier of R_Normed_Algebra_of_BoundedLinearOperators b1
for b5, b6 being Element of REAL holds
b2 + b3 = b3 + b2 &
(b2 + b3) + b4 = b2 + (b3 + b4) &
b2 + 0. R_Normed_Algebra_of_BoundedLinearOperators b1 = b2 &
(ex b7 being Element of the carrier of R_Normed_Algebra_of_BoundedLinearOperators b1 st
b2 + b7 = 0. R_Normed_Algebra_of_BoundedLinearOperators b1) &
(b2 * b3) * b4 = b2 * (b3 * b4) &
b2 * 1. R_Normed_Algebra_of_BoundedLinearOperators b1 = b2 &
(1. R_Normed_Algebra_of_BoundedLinearOperators b1) * b2 = b2 &
b2 * (b3 + b4) = (b2 * b3) + (b2 * b4) &
(b3 + b4) * b2 = (b3 * b2) + (b4 * b2) &
b5 * (b2 * b3) = (b5 * b2) * b3 &
(b5 * b6) * (b2 * b3) = (b5 * b2) * (b6 * b3) &
b5 * (b2 + b3) = (b5 * b2) + (b5 * b3) &
(b5 + b6) * b2 = (b5 * b2) + (b6 * b2) &
(b5 * b6) * b2 = b5 * (b6 * b2) &
1 * b2 = b2;
:: LOPBAN_2:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
R_Normed_Algebra_of_BoundedLinearOperators b1 is RealNormSpace-like & R_Normed_Algebra_of_BoundedLinearOperators b1 is Abelian & R_Normed_Algebra_of_BoundedLinearOperators b1 is add-associative & R_Normed_Algebra_of_BoundedLinearOperators b1 is right_zeroed & R_Normed_Algebra_of_BoundedLinearOperators b1 is right_complementable & R_Normed_Algebra_of_BoundedLinearOperators b1 is associative & R_Normed_Algebra_of_BoundedLinearOperators b1 is right_unital & R_Normed_Algebra_of_BoundedLinearOperators b1 is right-distributive & R_Normed_Algebra_of_BoundedLinearOperators b1 is Algebra-like & R_Normed_Algebra_of_BoundedLinearOperators b1 is RealLinearSpace-like;
:: LOPBAN_2:exreg 4
registration
cluster non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like Algebra-like RealNormSpace-like associative right-distributive right_unital strict Normed_AlgebraStr;
end;
:: LOPBAN_2:modenot 2
definition
mode Normed_Algebra is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like Algebra-like RealNormSpace-like associative right-distributive right_unital Normed_AlgebraStr;
end;
:: LOPBAN_2:funcreg 10
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
cluster R_Normed_Algebra_of_BoundedLinearOperators a1 -> right_complementable Abelian add-associative right_zeroed RealLinearSpace-like Algebra-like RealNormSpace-like associative right-distributive right_unital;
end;
:: LOPBAN_2:attrnot 2 => LOPBAN_2:attr 2
definition
let a1 be non empty Normed_AlgebraStr;
attr a1 is Banach_Algebra-like_1 means
for b1, b2 being Element of the carrier of a1 holds
||.b1 * b2.|| <= ||.b1.|| * ||.b2.||;
end;
:: LOPBAN_2:dfs 9
definiens
let a1 be non empty Normed_AlgebraStr;
To prove
a1 is Banach_Algebra-like_1
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1 holds
||.b1 * b2.|| <= ||.b1.|| * ||.b2.||;
:: LOPBAN_2:def 9
theorem
for b1 being non empty Normed_AlgebraStr holds
b1 is Banach_Algebra-like_1
iff
for b2, b3 being Element of the carrier of b1 holds
||.b2 * b3.|| <= ||.b2.|| * ||.b3.||;
:: LOPBAN_2:attrnot 3 => LOPBAN_2:attr 3
definition
let a1 be non empty Normed_AlgebraStr;
attr a1 is Banach_Algebra-like_2 means
||.1. a1.|| = 1;
end;
:: LOPBAN_2:dfs 10
definiens
let a1 be non empty Normed_AlgebraStr;
To prove
a1 is Banach_Algebra-like_2
it is sufficient to prove
thus ||.1. a1.|| = 1;
:: LOPBAN_2:def 10
theorem
for b1 being non empty Normed_AlgebraStr holds
b1 is Banach_Algebra-like_2
iff
||.1. b1.|| = 1;
:: LOPBAN_2:attrnot 4 => LOPBAN_2:attr 4
definition
let a1 be non empty Normed_AlgebraStr;
attr a1 is Banach_Algebra-like_3 means
for b1 being Element of REAL
for b2, b3 being Element of the carrier of a1 holds
b1 * (b2 * b3) = b2 * (b1 * b3);
end;
:: LOPBAN_2:dfs 11
definiens
let a1 be non empty Normed_AlgebraStr;
To prove
a1 is Banach_Algebra-like_3
it is sufficient to prove
thus for b1 being Element of REAL
for b2, b3 being Element of the carrier of a1 holds
b1 * (b2 * b3) = b2 * (b1 * b3);
:: LOPBAN_2:def 11
theorem
for b1 being non empty Normed_AlgebraStr holds
b1 is Banach_Algebra-like_3
iff
for b2 being Element of REAL
for b3, b4 being Element of the carrier of b1 holds
b2 * (b3 * b4) = b3 * (b2 * b4);
:: LOPBAN_2:attrnot 5 => LOPBAN_2:attr 5
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like Algebra-like RealNormSpace-like associative right-distributive right_unital Normed_AlgebraStr;
attr a1 is Banach_Algebra-like means
a1 is Banach_Algebra-like_1 & a1 is Banach_Algebra-like_2 & a1 is Banach_Algebra-like_3 & a1 is left_unital & a1 is left-distributive & a1 is complete;
end;
:: LOPBAN_2:dfs 12
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like Algebra-like RealNormSpace-like associative right-distributive right_unital Normed_AlgebraStr;
To prove
a1 is Banach_Algebra-like
it is sufficient to prove
thus a1 is Banach_Algebra-like_1 & a1 is Banach_Algebra-like_2 & a1 is Banach_Algebra-like_3 & a1 is left_unital & a1 is left-distributive & a1 is complete;
:: LOPBAN_2:def 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like Algebra-like RealNormSpace-like associative right-distributive right_unital Normed_AlgebraStr holds
b1 is Banach_Algebra-like
iff
b1 is Banach_Algebra-like_1 & b1 is Banach_Algebra-like_2 & b1 is Banach_Algebra-like_3 & b1 is left_unital & b1 is left-distributive & b1 is complete;
:: LOPBAN_2:condreg 1
registration
cluster non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like Algebra-like RealNormSpace-like associative right-distributive right_unital Banach_Algebra-like -> complete left-distributive left_unital Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 (Normed_AlgebraStr);
end;
:: LOPBAN_2:condreg 2
registration
cluster non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like Algebra-like RealNormSpace-like complete associative right-distributive left-distributive right_unital left_unital Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 -> Banach_Algebra-like (Normed_AlgebraStr);
end;
:: LOPBAN_2:funcreg 11
registration
let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR;
cluster R_Normed_Algebra_of_BoundedLinearOperators a1 -> Banach_Algebra-like;
end;
:: LOPBAN_2:exreg 5
registration
cluster non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed RealLinearSpace-like zeroed Algebra-like RealNormSpace-like associative right-distributive right_unital Banach_Algebra-like Normed_AlgebraStr;
end;
:: LOPBAN_2:modenot 3
definition
mode Banach_Algebra is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like Algebra-like RealNormSpace-like associative right-distributive right_unital Banach_Algebra-like Normed_AlgebraStr;
end;
:: LOPBAN_2:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
1. Ring_of_BoundedLinearOperators b1 = FuncUnit b1;
:: LOPBAN_2:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
1. R_Algebra_of_BoundedLinearOperators b1 = FuncUnit b1;
:: LOPBAN_2:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
1. R_Normed_Algebra_of_BoundedLinearOperators b1 = FuncUnit b1;