Article LOPBAN_5, MML version 4.99.1005

:: LOPBAN_5:th 1
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of REAL
      st b1 is bounded & 0 <= b2
   holds lim_inf (b2 (#) b1) = b2 * lim_inf b1;

:: LOPBAN_5:th 2
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of REAL
      st b1 is bounded & 0 <= b2
   holds lim_sup (b2 (#) b1) = b2 * lim_sup b1;

:: LOPBAN_5:funcreg 1
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR;
  cluster MetricSpaceNorm a1 -> non empty Reflexive discerning symmetric triangle complete;
end;

:: LOPBAN_5:funcnot 1 => LOPBAN_5:func 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR;
  let a2 be Element of the carrier of a1;
  let a3 be real set;
  func Ball(A2,A3) -> Element of bool the carrier of a1 equals
    {b1 where b1 is Element of the carrier of a1: ||.a2 - b1.|| < a3};
end;

:: LOPBAN_5:def 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR
for b2 being Element of the carrier of b1
for b3 being real set holds
   Ball(b2,b3) = {b4 where b4 is Element of the carrier of b1: ||.b2 - b4.|| < b3};

:: LOPBAN_5:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
      st union proj2 b2 = the carrier of b1 &
         (for b3 being Element of NAT holds
            b2 . b3 is closed(b1))
   holds ex b3 being Element of NAT st
      ex b4 being Element of REAL st
         ex b5 being Element of the carrier of b1 st
            0 < b4 & Ball(b5,b4) c= b2 . b3;

:: LOPBAN_5:th 4
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
   b3 is_Lipschitzian_on the carrier of b1 &
    b3 is_continuous_on the carrier of b1 &
    (for b4 being Element of the carrier of b1 holds
       b3 is_continuous_in b4);

:: LOPBAN_5:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR
for 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_NormSpace_of_BoundedLinearOperators(b1,b2)
      st for b4 being Element of the carrier of b1 holds
           ex b5 being real set st
              0 <= b5 &
               (for b6 being Element of the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b2)
                     st b6 in b3
                  holds ||.b6 . b4.|| <= b5)
   holds ex b4 being real set st
      0 <= b4 &
       (for b5 being Element of the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b2)
             st b5 in b3
          holds ||.b5.|| <= b4);

:: LOPBAN_5:funcnot 2 => LOPBAN_5:func 2
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 Relation of NAT,the carrier of R_NormSpace_of_BoundedLinearOperators(a1,a2);
  let a4 be Element of the carrier of a1;
  func A3 # A4 -> Function-like quasi_total Relation of NAT,the carrier of a2 means
    for b1 being Element of NAT holds
       it . b1 = (a3 . b1) . a4;
end;

:: LOPBAN_5:def 2
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 NAT,the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b2)
for b4 being Element of the carrier of b1
for b5 being Function-like quasi_total Relation of NAT,the carrier of b2 holds
      b5 = b3 # b4
   iff
      for b6 being Element of NAT holds
         b5 . b6 = (b3 . b6) . b4;

:: LOPBAN_5:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR
for 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 NAT,the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b2)
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st for b5 being Element of the carrier of b1 holds
           b3 # b5 is convergent(b2) & b4 . b5 = lim (b3 # b5)
   holds b4 is Function-like quasi_total additive homogeneous bounded Relation of the carrier of b1,the carrier of b2 &
    (for b5 being Element of the carrier of b1 holds
       ||.b4 . b5.|| <= (lim_inf ||.b3.||) * ||.b5.||) &
    (for b5 being Element of the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b2)
          st b5 = b4
       holds ||.b5.|| <= lim_inf ||.b3.||);

:: LOPBAN_5:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR
for b2 being Element of bool the carrier of LinearTopSpaceNorm b1
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR
for b4 being Function-like quasi_total Relation of NAT,the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b3)
   st b2 is dense(LinearTopSpaceNorm b1) &
      (for b5 being Element of the carrier of b1
            st b5 in b2
         holds b4 # b5 is convergent(b3)) &
      (for b5 being Element of the carrier of b1 holds
         ex b6 being real set st
            0 <= b6 &
             (for b7 being Element of NAT holds
                ||.(b4 # b5) . b7.|| <= b6))
for b5 being Element of the carrier of b1 holds
   b4 # b5 is convergent(b3);

:: LOPBAN_5:th 8
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR
for b3 being Element of bool the carrier of LinearTopSpaceNorm b1
for b4 being Function-like quasi_total Relation of NAT,the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b2)
      st b3 is dense(LinearTopSpaceNorm b1) &
         (for b5 being Element of the carrier of b1
               st b5 in b3
            holds b4 # b5 is convergent(b2)) &
         (for b5 being Element of the carrier of b1 holds
            ex b6 being real set st
               0 <= b6 &
                (for b7 being Element of NAT holds
                   ||.(b4 # b5) . b7.|| <= b6))
   holds ex b5 being Element of the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b2) st
      (for b6 being Element of the carrier of b1 holds
          b4 # b6 is convergent(b2) &
           b5 . b6 = lim (b4 # b6) &
           ||.b5 . b6.|| <= (lim_inf ||.b4.||) * ||.b6.||) &
       ||.b5.|| <= lim_inf ||.b4.||;