Article LOPBAN_1, MML version 4.99.1005

:: LOPBAN_1:funcnot 1 => LOPBAN_1:func 1
definition
  let a1 be set;
  let a2 be non empty set;
  let a3 be Function-like quasi_total Relation of [:REAL,a2:],a2;
  let a4 be real set;
  let a5 be Function-like quasi_total Relation of a1,a2;
  redefine func a3 [;](a4,a5) -> Element of Funcs(a1,a2);
end;

:: LOPBAN_1:th 1
theorem
for b1 being non empty set
for b2 being non empty addLoopStr holds
   ex b3 being Function-like quasi_total Relation of [:Funcs(b1,the carrier of b2),Funcs(b1,the carrier of b2):],Funcs(b1,the carrier of b2) st
      for b4, b5 being Element of Funcs(b1,the carrier of b2) holds
      b3 .(b4,b5) = (the addF of b2) .:(b4,b5);

:: LOPBAN_1:th 2
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   ex b3 being Function-like quasi_total Relation of [:REAL,Funcs(b1,the carrier of b2):],Funcs(b1,the carrier of b2) st
      for b4 being Element of REAL
      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);

:: LOPBAN_1:funcnot 2 => LOPBAN_1:func 2
definition
  let a1 be non empty set;
  let a2 be non empty addLoopStr;
  func FuncAdd(A1,A2) -> Function-like quasi_total Relation of [:Funcs(a1,the carrier of a2),Funcs(a1,the carrier of a2):],Funcs(a1,the carrier of a2) means
    for b1, b2 being Element of Funcs(a1,the carrier of a2) holds
    it .(b1,b2) = (the addF of a2) .:(b1,b2);
end;

:: LOPBAN_1:def 1
theorem
for b1 being non empty set
for b2 being non empty addLoopStr
for b3 being Function-like quasi_total Relation of [:Funcs(b1,the carrier of b2),Funcs(b1,the carrier of b2):],Funcs(b1,the carrier of b2) holds
      b3 = FuncAdd(b1,b2)
   iff
      for b4, b5 being Element of Funcs(b1,the carrier of b2) holds
      b3 .(b4,b5) = (the addF of b2) .:(b4,b5);

:: LOPBAN_1:funcnot 3 => LOPBAN_1:func 3
definition
  let a1 be non empty set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  func FuncExtMult(A1,A2) -> Function-like quasi_total Relation of [:REAL,Funcs(a1,the carrier of a2):],Funcs(a1,the carrier of a2) means
    for b1 being Element of REAL
    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;

:: LOPBAN_1:def 2
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3 being Function-like quasi_total Relation of [:REAL,Funcs(b1,the carrier of b2):],Funcs(b1,the carrier of b2) holds
      b3 = FuncExtMult(b1,b2)
   iff
      for b4 being Element of REAL
      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);

:: LOPBAN_1:funcnot 4 => LOPBAN_1:func 4
definition
  let a1 be set;
  let a2 be non empty ZeroStr;
  func FuncZero(A1,A2) -> Element of Funcs(a1,the carrier of a2) equals
    a1 --> 0. a2;
end;

:: LOPBAN_1:def 3
theorem
for b1 being set
for b2 being non empty ZeroStr holds
   FuncZero(b1,b2) = b1 --> 0. b2;

:: LOPBAN_1:th 3
theorem
for b1 being non empty set
for b2 being non empty addLoopStr
for b3, b4, b5 being Element of Funcs(b1,the carrier of b2) holds
   b5 = (FuncAdd(b1,b2)) .(b3,b4)
iff
   for b6 being Element of b1 holds
      b5 . b6 = (b3 . b6) + (b4 . b6);

:: LOPBAN_1:th 5
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3, b4 being Element of Funcs(b1,the carrier of b2)
for b5 being Element of REAL holds
      b3 = (FuncExtMult(b1,b2)) . [b5,b4]
   iff
      for b6 being Element of b1 holds
         b3 . b6 = b5 * (b4 . b6);

:: LOPBAN_1:th 6
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3, b4 being Element of Funcs(b1,the carrier of b2) holds
(FuncAdd(b1,b2)) .(b3,b4) = (FuncAdd(b1,b2)) .(b4,b3);

:: LOPBAN_1:th 7
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
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);

:: LOPBAN_1:th 8
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3 being Element of Funcs(b1,the carrier of b2) holds
   (FuncAdd(b1,b2)) .(FuncZero(b1,b2),b3) = b3;

:: LOPBAN_1:th 9
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3 being Element of Funcs(b1,the carrier of b2) holds
   (FuncAdd(b1,b2)) .(b3,(FuncExtMult(b1,b2)) . [- 1,b3]) = FuncZero(b1,b2);

:: LOPBAN_1:th 10
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3 being Element of Funcs(b1,the carrier of b2) holds
   (FuncExtMult(b1,b2)) . [1,b3] = b3;

:: LOPBAN_1:th 11
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3 being Element of Funcs(b1,the carrier of b2)
for b4, b5 being Element of REAL holds
(FuncExtMult(b1,b2)) . [b4,(FuncExtMult(b1,b2)) . [b5,b3]] = (FuncExtMult(b1,b2)) . [b4 * b5,b3];

:: LOPBAN_1:th 12
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3 being Element of Funcs(b1,the carrier of b2)
for b4, b5 being Element of REAL holds
(FuncAdd(b1,b2)) .((FuncExtMult(b1,b2)) . [b4,b3],(FuncExtMult(b1,b2)) . [b5,b3]) = (FuncExtMult(b1,b2)) . [b4 + b5,b3];

:: LOPBAN_1:th 13
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   RLSStruct(#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 RealLinearSpace-like RLSStruct;

:: LOPBAN_1:funcnot 5 => LOPBAN_1:func 5
definition
  let a1 be non empty set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  func RealVectSpace(A1,A2) -> non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct equals
    RLSStruct(#Funcs(a1,the carrier of a2),FuncZero(a1,a2),FuncAdd(a1,a2),FuncExtMult(a1,a2)#);
end;

:: LOPBAN_1:def 4
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   RealVectSpace(b1,b2) = RLSStruct(#Funcs(b1,the carrier of b2),FuncZero(b1,b2),FuncAdd(b1,b2),FuncExtMult(b1,b2)#);

:: LOPBAN_1:funcreg 1
registration
  let a1 be non empty set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  cluster RealVectSpace(a1,a2) -> non empty right_complementable strict Abelian add-associative right_zeroed RealLinearSpace-like;
end;

:: LOPBAN_1:funcreg 2
registration
  let a1 be non empty set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  cluster RealVectSpace(a1,a2) -> non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like constituted-Functions;
end;

:: LOPBAN_1:funcnot 6 => LOPBAN_1:func 6
definition
  let a1 be non empty set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a3 be Element of the carrier of RealVectSpace(a1,a2);
  let a4 be Element of a1;
  redefine func a3 . a4 -> Element of the carrier of a2;
end;

:: LOPBAN_1:th 14
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3, b4, b5 being Element of the carrier of RealVectSpace(b1,b2) holds
   b5 = b3 + b4
iff
   for b6 being Element of b1 holds
      b5 . b6 = (b3 . b6) + (b4 . b6);

:: LOPBAN_1:th 15
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3, b4 being Element of the carrier of RealVectSpace(b1,b2)
for b5 being Element of REAL holds
      b4 = b5 * b3
   iff
      for b6 being Element of b1 holds
         b4 . b6 = b5 * (b3 . b6);

:: LOPBAN_1:th 16
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   0. RealVectSpace(b1,b2) = b1 --> 0. b2;

:: LOPBAN_1:attrnot 1 => LOPBAN_1:attr 1
definition
  let a1 be non empty RLSStruct;
  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;

:: LOPBAN_1:dfs 5
definiens
  let a1 be non empty RLSStruct;
  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);

:: LOPBAN_1:def 5
theorem
for b1 being non empty RLSStruct
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);

:: LOPBAN_1:attrnot 2 => LOPBAN_1:attr 2
definition
  let a1, a2 be non empty RLSStruct;
  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 REAL holds
       a3 . (b2 * b1) = b2 * (a3 . b1);
end;

:: LOPBAN_1:dfs 6
definiens
  let a1, a2 be non empty RLSStruct;
  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 REAL holds
       a3 . (b2 * b1) = b2 * (a3 . b1);

:: LOPBAN_1:def 6
theorem
for b1, b2 being non empty RLSStruct
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 REAL holds
         b3 . (b5 * b4) = b5 * (b3 . b4);

:: LOPBAN_1:exreg 1
registration
  let a1 be non empty RLSStruct;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  cluster non empty Relation-like Function-like quasi_total total additive homogeneous Relation of the carrier of a1,the carrier of a2;
end;

:: LOPBAN_1:modenot 1
definition
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  mode LinearOperator of a1,a2 is Function-like quasi_total additive homogeneous Relation of the carrier of a1,the carrier of a2;
end;

:: LOPBAN_1:funcnot 7 => LOPBAN_1:func 7
definition
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  func LinearOperators(A1,A2) -> Element of bool the carrier of RealVectSpace(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;

:: LOPBAN_1:def 7
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3 being Element of bool the carrier of RealVectSpace(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;

:: LOPBAN_1:funcreg 3
registration
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  cluster LinearOperators(a1,a2) -> non empty functional;
end;

:: LOPBAN_1:th 17
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
LinearOperators(b1,b2) is linearly-closed(RealVectSpace(the carrier of b1,b2));

:: LOPBAN_1:th 18
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
RLSStruct(#LinearOperators(b1,b2),Zero_(LinearOperators(b1,b2),RealVectSpace(the carrier of b1,b2)),Add_(LinearOperators(b1,b2),RealVectSpace(the carrier of b1,b2)),Mult_(LinearOperators(b1,b2),RealVectSpace(the carrier of b1,b2))#) is Subspace of RealVectSpace(the carrier of b1,b2);

:: LOPBAN_1:funcreg 4
registration
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  cluster RLSStruct(#LinearOperators(a1,a2),Zero_(LinearOperators(a1,a2),RealVectSpace(the carrier of a1,a2)),Add_(LinearOperators(a1,a2),RealVectSpace(the carrier of a1,a2)),Mult_(LinearOperators(a1,a2),RealVectSpace(the carrier of a1,a2))#) -> right_complementable strict Abelian add-associative right_zeroed RealLinearSpace-like;
end;

:: LOPBAN_1:funcnot 8 => LOPBAN_1:func 8
definition
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  func R_VectorSpace_of_LinearOperators(A1,A2) -> non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct equals
    RLSStruct(#LinearOperators(a1,a2),Zero_(LinearOperators(a1,a2),RealVectSpace(the carrier of a1,a2)),Add_(LinearOperators(a1,a2),RealVectSpace(the carrier of a1,a2)),Mult_(LinearOperators(a1,a2),RealVectSpace(the carrier of a1,a2))#);
end;

:: LOPBAN_1:def 8
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
R_VectorSpace_of_LinearOperators(b1,b2) = RLSStruct(#LinearOperators(b1,b2),Zero_(LinearOperators(b1,b2),RealVectSpace(the carrier of b1,b2)),Add_(LinearOperators(b1,b2),RealVectSpace(the carrier of b1,b2)),Mult_(LinearOperators(b1,b2),RealVectSpace(the carrier of b1,b2))#);

:: LOPBAN_1:funcreg 5
registration
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  cluster R_VectorSpace_of_LinearOperators(a1,a2) -> non empty right_complementable strict Abelian add-associative right_zeroed RealLinearSpace-like;
end;

:: LOPBAN_1:funcreg 6
registration
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  cluster R_VectorSpace_of_LinearOperators(a1,a2) -> non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like constituted-Functions;
end;

:: LOPBAN_1:funcnot 9 => LOPBAN_1:func 9
definition
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a3 be Element of the carrier of R_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;

:: LOPBAN_1:th 20
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3, b4, b5 being Element of the carrier of R_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);

:: LOPBAN_1:th 21
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3, b4 being Element of the carrier of R_VectorSpace_of_LinearOperators(b1,b2)
for b5 being Element of REAL holds
      b4 = b5 * b3
   iff
      for b6 being Element of the carrier of b1 holds
         b4 . b6 = b5 * (b3 . b6);

:: LOPBAN_1:th 22
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
0. R_VectorSpace_of_LinearOperators(b1,b2) = (the carrier of b1) --> 0. b2;

:: LOPBAN_1:th 23
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
(the carrier of b1) --> 0. b2 is Function-like quasi_total additive homogeneous Relation of the carrier of b1,the carrier of b2;

:: LOPBAN_1:th 24
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 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.||;

:: LOPBAN_1:attrnot 3 => LOPBAN_1:attr 3
definition
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  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;

:: LOPBAN_1:dfs 9
definiens
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  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.||);

:: LOPBAN_1:def 9
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
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.||);

:: LOPBAN_1:th 25
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
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);

:: LOPBAN_1:exreg 2
registration
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  cluster non empty Relation-like Function-like quasi_total total additive homogeneous bounded Relation of the carrier of a1,the carrier of a2;
end;

:: LOPBAN_1:funcnot 10 => LOPBAN_1:func 10
definition
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  func BoundedLinearOperators(A1,A2) -> Element of bool the carrier of R_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;

:: LOPBAN_1:def 10
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Element of bool the carrier of R_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;

:: LOPBAN_1:funcreg 7
registration
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  cluster BoundedLinearOperators(a1,a2) -> non empty;
end;

:: LOPBAN_1:th 26
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
BoundedLinearOperators(b1,b2) is linearly-closed(R_VectorSpace_of_LinearOperators(b1,b2));

:: LOPBAN_1:th 27
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
RLSStruct(#BoundedLinearOperators(b1,b2),Zero_(BoundedLinearOperators(b1,b2),R_VectorSpace_of_LinearOperators(b1,b2)),Add_(BoundedLinearOperators(b1,b2),R_VectorSpace_of_LinearOperators(b1,b2)),Mult_(BoundedLinearOperators(b1,b2),R_VectorSpace_of_LinearOperators(b1,b2))#) is Subspace of R_VectorSpace_of_LinearOperators(b1,b2);

:: LOPBAN_1:funcreg 8
registration
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  cluster RLSStruct(#BoundedLinearOperators(a1,a2),Zero_(BoundedLinearOperators(a1,a2),R_VectorSpace_of_LinearOperators(a1,a2)),Add_(BoundedLinearOperators(a1,a2),R_VectorSpace_of_LinearOperators(a1,a2)),Mult_(BoundedLinearOperators(a1,a2),R_VectorSpace_of_LinearOperators(a1,a2))#) -> right_complementable strict Abelian add-associative right_zeroed RealLinearSpace-like;
end;

:: LOPBAN_1:funcnot 11 => LOPBAN_1:func 11
definition
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  func R_VectorSpace_of_BoundedLinearOperators(A1,A2) -> non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct equals
    RLSStruct(#BoundedLinearOperators(a1,a2),Zero_(BoundedLinearOperators(a1,a2),R_VectorSpace_of_LinearOperators(a1,a2)),Add_(BoundedLinearOperators(a1,a2),R_VectorSpace_of_LinearOperators(a1,a2)),Mult_(BoundedLinearOperators(a1,a2),R_VectorSpace_of_LinearOperators(a1,a2))#);
end;

:: LOPBAN_1:def 11
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
R_VectorSpace_of_BoundedLinearOperators(b1,b2) = RLSStruct(#BoundedLinearOperators(b1,b2),Zero_(BoundedLinearOperators(b1,b2),R_VectorSpace_of_LinearOperators(b1,b2)),Add_(BoundedLinearOperators(b1,b2),R_VectorSpace_of_LinearOperators(b1,b2)),Mult_(BoundedLinearOperators(b1,b2),R_VectorSpace_of_LinearOperators(b1,b2))#);

:: LOPBAN_1:funcreg 9
registration
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  cluster R_VectorSpace_of_BoundedLinearOperators(a1,a2) -> non empty right_complementable strict Abelian add-associative right_zeroed RealLinearSpace-like;
end;

:: LOPBAN_1:condreg 1
registration
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  cluster -> Relation-like Function-like (Element of the carrier of R_VectorSpace_of_BoundedLinearOperators(a1,a2));
end;

:: LOPBAN_1:funcnot 12 => LOPBAN_1:func 12
definition
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a3 be Element of the carrier of R_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;

:: LOPBAN_1:th 29
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4, b5 being Element of the carrier of R_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);

:: LOPBAN_1:th 30
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Element of the carrier of R_VectorSpace_of_BoundedLinearOperators(b1,b2)
for b5 being Element of REAL holds
      b4 = b5 * b3
   iff
      for b6 being Element of the carrier of b1 holds
         b4 . b6 = b5 * (b3 . b6);

:: LOPBAN_1:th 31
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
0. R_VectorSpace_of_BoundedLinearOperators(b1,b2) = (the carrier of b1) --> 0. b2;

:: LOPBAN_1:funcnot 13 => LOPBAN_1:func 13
definition
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  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;

:: LOPBAN_1:def 12
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
      st b3 in BoundedLinearOperators(b1,b2)
   holds modetrans(b3,b1,b2) = b3;

:: LOPBAN_1:funcnot 14 => LOPBAN_1:func 14
definition
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  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;

:: LOPBAN_1:def 13
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
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};

:: LOPBAN_1:th 32
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
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;

:: LOPBAN_1:th 33
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
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;

:: LOPBAN_1:th 34
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR 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);

:: LOPBAN_1:funcnot 15 => LOPBAN_1:func 15
definition
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  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;

:: LOPBAN_1:def 14
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
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);

:: LOPBAN_1:th 35
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
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;

:: LOPBAN_1:th 36
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
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;

:: LOPBAN_1:funcnot 16 => LOPBAN_1:func 16
definition
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  func R_NormSpace_of_BoundedLinearOperators(A1,A2) -> non empty NORMSTR equals
    NORMSTR(#BoundedLinearOperators(a1,a2),Zero_(BoundedLinearOperators(a1,a2),R_VectorSpace_of_LinearOperators(a1,a2)),Add_(BoundedLinearOperators(a1,a2),R_VectorSpace_of_LinearOperators(a1,a2)),Mult_(BoundedLinearOperators(a1,a2),R_VectorSpace_of_LinearOperators(a1,a2)),BoundedLinearOperatorsNorm(a1,a2)#);
end;

:: LOPBAN_1:def 15
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
R_NormSpace_of_BoundedLinearOperators(b1,b2) = NORMSTR(#BoundedLinearOperators(b1,b2),Zero_(BoundedLinearOperators(b1,b2),R_VectorSpace_of_LinearOperators(b1,b2)),Add_(BoundedLinearOperators(b1,b2),R_VectorSpace_of_LinearOperators(b1,b2)),Mult_(BoundedLinearOperators(b1,b2),R_VectorSpace_of_LinearOperators(b1,b2)),BoundedLinearOperatorsNorm(b1,b2)#);

:: LOPBAN_1:th 37
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
(the carrier of b1) --> 0. b2 = 0. R_NormSpace_of_BoundedLinearOperators(b1,b2);

:: LOPBAN_1:th 38
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Element of the carrier of R_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.||;

:: LOPBAN_1:th 39
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Element of the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b2) holds
   0 <= ||.b3.||;

:: LOPBAN_1:th 40
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Element of the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b2)
      st b3 = 0. R_NormSpace_of_BoundedLinearOperators(b1,b2)
   holds 0 = ||.b3.||;

:: LOPBAN_1:condreg 2
registration
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  cluster -> Relation-like Function-like (Element of the carrier of R_NormSpace_of_BoundedLinearOperators(a1,a2));
end;

:: LOPBAN_1:funcnot 17 => LOPBAN_1:func 17
definition
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a3 be Element of the carrier of R_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;

:: LOPBAN_1:th 41
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4, b5 being Element of the carrier of R_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);

:: LOPBAN_1:th 42
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Element of the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b2)
for b5 being Element of REAL holds
      b4 = b5 * b3
   iff
      for b6 being Element of the carrier of b1 holds
         b4 . b6 = b5 * (b3 . b6);

:: LOPBAN_1:th 43
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Element of the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b2)
for b5 being Element of REAL holds
   (||.b3.|| = 0 implies b3 = 0. R_NormSpace_of_BoundedLinearOperators(b1,b2)) &
    (b3 = 0. R_NormSpace_of_BoundedLinearOperators(b1,b2) implies ||.b3.|| = 0) &
    ||.b5 * b3.|| = (abs b5) * ||.b3.|| &
    ||.b3 + b4.|| <= ||.b3.|| + ||.b4.||;

:: LOPBAN_1:th 44
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
R_NormSpace_of_BoundedLinearOperators(b1,b2) is RealNormSpace-like;

:: LOPBAN_1:th 45
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
R_NormSpace_of_BoundedLinearOperators(b1,b2) is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;

:: LOPBAN_1:funcreg 10
registration
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  cluster R_NormSpace_of_BoundedLinearOperators(a1,a2) -> non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like;
end;

:: LOPBAN_1:th 46
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4, b5 being Element of the carrier of R_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);

:: LOPBAN_1:attrnot 4 => LOPBAN_1:attr 4
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  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;

:: LOPBAN_1:dfs 16
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
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);

:: LOPBAN_1:def 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR 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);

:: LOPBAN_1:exreg 3
registration
  cluster non empty left_complementable right_complementable Abelian add-associative right_zeroed RealLinearSpace-like zeroed RealNormSpace-like complete NORMSTR;
end;

:: LOPBAN_1:modenot 2
definition
  mode RealBanachSpace is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR;
end;

:: LOPBAN_1:th 47
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 NAT,the carrier of b1
      st b2 is convergent(b1)
   holds ||.b2.|| is convergent &
    lim ||.b2.|| = ||.lim b2.||;

:: LOPBAN_1:th 48
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
   st b2 is complete
for b3 being Function-like quasi_total Relation of NAT,the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b2)
      st b3 is CCauchy(R_NormSpace_of_BoundedLinearOperators(b1,b2))
   holds b3 is convergent(R_NormSpace_of_BoundedLinearOperators(b1,b2));

:: LOPBAN_1:th 49
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR holds
   R_NormSpace_of_BoundedLinearOperators(b1,b2) is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR;

:: LOPBAN_1:funcreg 11
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR;
  cluster R_NormSpace_of_BoundedLinearOperators(a1,a2) -> non empty complete;
end;