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(AlgebraStrNORMSTR) 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;