Article CSSPACE4, MML version 4.99.1005

:: CSSPACE4:funcnot 1 => CSSPACE4:func 1
definition
  func the_set_of_BoundedComplexSequences -> Element of bool the carrier of Linear_Space_of_ComplexSequences means
    for b1 being set holds
          b1 in it
       iff
          b1 in the_set_of_ComplexSequences & seq_id b1 is bounded;
end;

:: CSSPACE4:def 1
theorem
for b1 being Element of bool the carrier of Linear_Space_of_ComplexSequences holds
      b1 = the_set_of_BoundedComplexSequences
   iff
      for b2 being set holds
            b2 in b1
         iff
            b2 in the_set_of_ComplexSequences & seq_id b2 is bounded;

:: CSSPACE4:funcreg 1
registration
  cluster the_set_of_BoundedComplexSequences -> non empty;
end;

:: CSSPACE4:funcreg 2
registration
  cluster the_set_of_BoundedComplexSequences -> linearly-closed;
end;

:: CSSPACE4:funcreg 3
registration
  cluster CLSStruct(#the_set_of_BoundedComplexSequences,Zero_(the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences),Add_(the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences),Mult_(the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences)#) -> right_complementable Abelian add-associative right_zeroed strict ComplexLinearSpace-like;
end;

:: CSSPACE4:funcnot 2 => CSSPACE4:func 2
definition
  func Complex_linfty_norm -> Function-like quasi_total Relation of the_set_of_BoundedComplexSequences,REAL means
    for b1 being set
          st b1 in the_set_of_BoundedComplexSequences
       holds it . b1 = sup rng |.seq_id b1.|;
end;

:: CSSPACE4:def 2
theorem
for b1 being Function-like quasi_total Relation of the_set_of_BoundedComplexSequences,REAL holds
      b1 = Complex_linfty_norm
   iff
      for b2 being set
            st b2 in the_set_of_BoundedComplexSequences
         holds b1 . b2 = sup rng |.seq_id b2.|;

:: CSSPACE4:th 2
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
      b1 is bounded & sup rng |.b1.| = 0
   iff
      for b2 being Element of NAT holds
         b1 . b2 = 0c;

:: CSSPACE4:funcreg 4
registration
  cluster CNORMSTR(#the_set_of_BoundedComplexSequences,Zero_(the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences),Add_(the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences),Mult_(the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences),Complex_linfty_norm#) -> right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like strict;
end;

:: CSSPACE4:funcnot 3 => CSSPACE4:func 3
definition
  func Complex_linfty_Space -> non empty CNORMSTR equals
    CNORMSTR(#the_set_of_BoundedComplexSequences,Zero_(the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences),Add_(the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences),Mult_(the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences),Complex_linfty_norm#);
end;

:: CSSPACE4:def 3
theorem
Complex_linfty_Space = CNORMSTR(#the_set_of_BoundedComplexSequences,Zero_(the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences),Add_(the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences),Mult_(the_set_of_BoundedComplexSequences,Linear_Space_of_ComplexSequences),Complex_linfty_norm#);

:: CSSPACE4:th 3
theorem
the carrier of Complex_linfty_Space = the_set_of_BoundedComplexSequences &
 (for b1 being set holds
       b1 is Element of the carrier of Complex_linfty_Space
    iff
       b1 is Function-like quasi_total Relation of NAT,COMPLEX & seq_id b1 is bounded) &
 0. Complex_linfty_Space = CZeroseq &
 (for b1 being Element of the carrier of Complex_linfty_Space holds
    b1 = seq_id b1) &
 (for b1, b2 being Element of the carrier of Complex_linfty_Space holds
 b1 + b2 = (seq_id b1) + seq_id b2) &
 (for b1 being Element of COMPLEX
 for b2 being Element of the carrier of Complex_linfty_Space holds
    b1 * b2 = b1 (#) seq_id b2) &
 (for b1 being Element of the carrier of Complex_linfty_Space holds
    - b1 = - seq_id b1 & seq_id - b1 = - seq_id b1) &
 (for b1, b2 being Element of the carrier of Complex_linfty_Space holds
 b1 - b2 = (seq_id b1) - seq_id b2) &
 (for b1 being Element of the carrier of Complex_linfty_Space holds
    seq_id b1 is bounded) &
 (for b1 being Element of the carrier of Complex_linfty_Space holds
    ||.b1.|| = sup rng |.seq_id b1.|);

:: CSSPACE4:th 4
theorem
for b1, b2 being Element of the carrier of Complex_linfty_Space
for b3 being Element of COMPLEX holds
   (||.b1.|| = 0 implies b1 = 0. Complex_linfty_Space) &
    (b1 = 0. Complex_linfty_Space implies ||.b1.|| = 0) &
    0 <= ||.b1.|| &
    ||.b1 + b2.|| <= ||.b1.|| + ||.b2.|| &
    ||.b3 * b1.|| = |.b3.| * ||.b1.||;

:: CSSPACE4:funcreg 5
registration
  cluster Complex_linfty_Space -> non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like;
end;

:: CSSPACE4:th 5
theorem
for b1 being Function-like quasi_total Relation of NAT,the carrier of Complex_linfty_Space
      st b1 is CCauchy(Complex_linfty_Space)
   holds b1 is convergent(Complex_linfty_Space);

:: CSSPACE4:th 6
theorem
Complex_linfty_Space is non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like complete CNORMSTR;

:: CSSPACE4:attrnot 1 => CSSPACE4:attr 1
definition
  let a1 be non empty set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  let a3 be Function-like quasi_total Relation 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 a1 holds
           ||.a3 . b2.|| <= b1);
end;

:: CSSPACE4:dfs 4
definiens
  let a1 be non empty set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  let a3 be Function-like quasi_total Relation 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 a1 holds
           ||.a3 . b2.|| <= b1);

:: CSSPACE4:def 4
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like quasi_total Relation of 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 b1 holds
             ||.b3 . b5.|| <= b4);

:: CSSPACE4:th 7
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like quasi_total Relation of b1,the carrier of b2
      st for b4 being Element of b1 holds
           b3 . b4 = 0. b2
   holds b3 is bounded(b1, b2);

:: CSSPACE4:exreg 1
registration
  let a1 be non empty set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  cluster Relation-like Function-like non empty total quasi_total bounded Relation of a1,the carrier of a2;
end;

:: CSSPACE4:funcnot 4 => CSSPACE4:func 4
definition
  let a1 be non empty set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  func ComplexBoundedFunctions(A1,A2) -> Element of bool the carrier of ComplexVectSpace(a1,a2) means
    for b1 being set holds
          b1 in it
       iff
          b1 is Function-like quasi_total bounded Relation of a1,the carrier of a2;
end;

:: CSSPACE4:def 5
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Element of bool the carrier of ComplexVectSpace(b1,b2) holds
      b3 = ComplexBoundedFunctions(b1,b2)
   iff
      for b4 being set holds
            b4 in b3
         iff
            b4 is Function-like quasi_total bounded Relation of b1,the carrier of b2;

:: CSSPACE4:funcreg 6
registration
  let a1 be non empty set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  cluster ComplexBoundedFunctions(a1,a2) -> non empty;
end;

:: CSSPACE4:th 8
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR holds
   ComplexBoundedFunctions(b1,b2) is linearly-closed(ComplexVectSpace(b1,b2));

:: CSSPACE4:th 9
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR holds
   CLSStruct(#ComplexBoundedFunctions(b1,b2),Zero_(ComplexBoundedFunctions(b1,b2),ComplexVectSpace(b1,b2)),Add_(ComplexBoundedFunctions(b1,b2),ComplexVectSpace(b1,b2)),Mult_(ComplexBoundedFunctions(b1,b2),ComplexVectSpace(b1,b2))#) is Subspace of ComplexVectSpace(b1,b2);

:: CSSPACE4:funcreg 7
registration
  let a1 be non empty set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  cluster CLSStruct(#ComplexBoundedFunctions(a1,a2),Zero_(ComplexBoundedFunctions(a1,a2),ComplexVectSpace(a1,a2)),Add_(ComplexBoundedFunctions(a1,a2),ComplexVectSpace(a1,a2)),Mult_(ComplexBoundedFunctions(a1,a2),ComplexVectSpace(a1,a2))#) -> right_complementable Abelian add-associative right_zeroed strict ComplexLinearSpace-like;
end;

:: CSSPACE4:funcnot 5 => CSSPACE4:func 5
definition
  let a1 be non empty set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  func C_VectorSpace_of_BoundedFunctions(A1,A2) -> non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct equals
    CLSStruct(#ComplexBoundedFunctions(a1,a2),Zero_(ComplexBoundedFunctions(a1,a2),ComplexVectSpace(a1,a2)),Add_(ComplexBoundedFunctions(a1,a2),ComplexVectSpace(a1,a2)),Mult_(ComplexBoundedFunctions(a1,a2),ComplexVectSpace(a1,a2))#);
end;

:: CSSPACE4:def 6
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR holds
   C_VectorSpace_of_BoundedFunctions(b1,b2) = CLSStruct(#ComplexBoundedFunctions(b1,b2),Zero_(ComplexBoundedFunctions(b1,b2),ComplexVectSpace(b1,b2)),Add_(ComplexBoundedFunctions(b1,b2),ComplexVectSpace(b1,b2)),Mult_(ComplexBoundedFunctions(b1,b2),ComplexVectSpace(b1,b2))#);

:: CSSPACE4:funcreg 8
registration
  let a1 be non empty set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  cluster C_VectorSpace_of_BoundedFunctions(a1,a2) -> non empty right_complementable Abelian add-associative right_zeroed strict ComplexLinearSpace-like;
end;

:: CSSPACE4:th 11
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3, b4, b5 being Element of the carrier of C_VectorSpace_of_BoundedFunctions(b1,b2)
for b6, b7, b8 being Function-like quasi_total bounded Relation of b1,the carrier of b2
      st b6 = b3 & b7 = b4 & b8 = b5
   holds    b5 = b3 + b4
   iff
      for b9 being Element of b1 holds
         b8 . b9 = (b6 . b9) + (b7 . b9);

:: CSSPACE4:th 12
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3, b4 being Element of the carrier of C_VectorSpace_of_BoundedFunctions(b1,b2)
for b5, b6 being Function-like quasi_total bounded Relation of b1,the carrier of b2
   st b5 = b3 & b6 = b4
for b7 being Element of COMPLEX holds
      b4 = b7 * b3
   iff
      for b8 being Element of b1 holds
         b6 . b8 = b7 * (b5 . b8);

:: CSSPACE4:th 13
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR holds
   0. C_VectorSpace_of_BoundedFunctions(b1,b2) = b1 --> 0. b2;

:: CSSPACE4:funcnot 6 => CSSPACE4:func 6
definition
  let a1 be non empty set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  let a3 be set;
  assume a3 in ComplexBoundedFunctions(a1,a2);
  func modetrans(A3,A1,A2) -> Function-like quasi_total bounded Relation of a1,the carrier of a2 equals
    a3;
end;

:: CSSPACE4:def 7
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being set
      st b3 in ComplexBoundedFunctions(b1,b2)
   holds modetrans(b3,b1,b2) = b3;

:: CSSPACE4:funcnot 7 => CSSPACE4:func 7
definition
  let a1 be non empty set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  let a3 be Function-like quasi_total Relation of a1,the carrier of a2;
  func PreNorms A3 -> non empty Element of bool REAL equals
    {||.a3 . b1.|| where b1 is Element of a1: TRUE};
end;

:: CSSPACE4:def 8
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like quasi_total Relation of b1,the carrier of b2 holds
   PreNorms b3 = {||.b3 . b4.|| where b4 is Element of b1: TRUE};

:: CSSPACE4:th 14
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like quasi_total bounded Relation of b1,the carrier of b2 holds
   PreNorms b3 is not empty & PreNorms b3 is bounded_above;

:: CSSPACE4:th 15
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like quasi_total Relation of b1,the carrier of b2 holds
      b3 is bounded(b1, b2)
   iff
      PreNorms b3 is bounded_above;

:: CSSPACE4:th 16
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR holds
   ex b3 being Function-like quasi_total Relation of ComplexBoundedFunctions(b1,b2),REAL st
      for b4 being set
            st b4 in ComplexBoundedFunctions(b1,b2)
         holds b3 . b4 = sup PreNorms modetrans(b4,b1,b2);

:: CSSPACE4:funcnot 8 => CSSPACE4:func 8
definition
  let a1 be non empty set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  func ComplexBoundedFunctionsNorm(A1,A2) -> Function-like quasi_total Relation of ComplexBoundedFunctions(a1,a2),REAL means
    for b1 being set
          st b1 in ComplexBoundedFunctions(a1,a2)
       holds it . b1 = sup PreNorms modetrans(b1,a1,a2);
end;

:: CSSPACE4:def 9
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like quasi_total Relation of ComplexBoundedFunctions(b1,b2),REAL holds
      b3 = ComplexBoundedFunctionsNorm(b1,b2)
   iff
      for b4 being set
            st b4 in ComplexBoundedFunctions(b1,b2)
         holds b3 . b4 = sup PreNorms modetrans(b4,b1,b2);

:: CSSPACE4:th 17
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like quasi_total bounded Relation of b1,the carrier of b2 holds
   modetrans(b3,b1,b2) = b3;

:: CSSPACE4:th 18
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like quasi_total bounded Relation of b1,the carrier of b2 holds
   (ComplexBoundedFunctionsNorm(b1,b2)) . b3 = sup PreNorms b3;

:: CSSPACE4:funcnot 9 => CSSPACE4:func 9
definition
  let a1 be non empty set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  func C_NormSpace_of_BoundedFunctions(A1,A2) -> non empty CNORMSTR equals
    CNORMSTR(#ComplexBoundedFunctions(a1,a2),Zero_(ComplexBoundedFunctions(a1,a2),ComplexVectSpace(a1,a2)),Add_(ComplexBoundedFunctions(a1,a2),ComplexVectSpace(a1,a2)),Mult_(ComplexBoundedFunctions(a1,a2),ComplexVectSpace(a1,a2)),ComplexBoundedFunctionsNorm(a1,a2)#);
end;

:: CSSPACE4:def 10
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR holds
   C_NormSpace_of_BoundedFunctions(b1,b2) = CNORMSTR(#ComplexBoundedFunctions(b1,b2),Zero_(ComplexBoundedFunctions(b1,b2),ComplexVectSpace(b1,b2)),Add_(ComplexBoundedFunctions(b1,b2),ComplexVectSpace(b1,b2)),Mult_(ComplexBoundedFunctions(b1,b2),ComplexVectSpace(b1,b2)),ComplexBoundedFunctionsNorm(b1,b2)#);

:: CSSPACE4:th 19
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR holds
   b1 --> 0. b2 = 0. C_NormSpace_of_BoundedFunctions(b1,b2);

:: CSSPACE4:th 20
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Element of the carrier of C_NormSpace_of_BoundedFunctions(b1,b2)
for b4 being Function-like quasi_total bounded Relation of b1,the carrier of b2
   st b4 = b3
for b5 being Element of b1 holds
   ||.b4 . b5.|| <= ||.b3.||;

:: CSSPACE4:th 21
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Element of the carrier of C_NormSpace_of_BoundedFunctions(b1,b2) holds
   0 <= ||.b3.||;

:: CSSPACE4:th 22
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Element of the carrier of C_NormSpace_of_BoundedFunctions(b1,b2)
      st b3 = 0. C_NormSpace_of_BoundedFunctions(b1,b2)
   holds 0 = ||.b3.||;

:: CSSPACE4:th 23
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3, b4, b5 being Element of the carrier of C_NormSpace_of_BoundedFunctions(b1,b2)
for b6, b7, b8 being Function-like quasi_total bounded Relation of b1,the carrier of b2
      st b6 = b3 & b7 = b4 & b8 = b5
   holds    b5 = b3 + b4
   iff
      for b9 being Element of b1 holds
         b8 . b9 = (b6 . b9) + (b7 . b9);

:: CSSPACE4:th 24
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3, b4 being Element of the carrier of C_NormSpace_of_BoundedFunctions(b1,b2)
for b5, b6 being Function-like quasi_total bounded Relation of b1,the carrier of b2
   st b5 = b3 & b6 = b4
for b7 being Element of COMPLEX holds
      b4 = b7 * b3
   iff
      for b8 being Element of b1 holds
         b6 . b8 = b7 * (b5 . b8);

:: CSSPACE4:th 25
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3, b4 being Element of the carrier of C_NormSpace_of_BoundedFunctions(b1,b2)
for b5 being Element of COMPLEX holds
   (||.b3.|| = 0 implies b3 = 0. C_NormSpace_of_BoundedFunctions(b1,b2)) &
    (b3 = 0. C_NormSpace_of_BoundedFunctions(b1,b2) implies ||.b3.|| = 0) &
    ||.b5 * b3.|| = |.b5.| * ||.b3.|| &
    ||.b3 + b4.|| <= ||.b3.|| + ||.b4.||;

:: CSSPACE4:th 26
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR holds
   C_NormSpace_of_BoundedFunctions(b1,b2) is ComplexNormSpace-like;

:: CSSPACE4:th 27
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR holds
   C_NormSpace_of_BoundedFunctions(b1,b2) is non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;

:: CSSPACE4:funcreg 9
registration
  let a1 be non empty set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  cluster C_NormSpace_of_BoundedFunctions(a1,a2) -> non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like;
end;

:: CSSPACE4:th 28
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3, b4, b5 being Element of the carrier of C_NormSpace_of_BoundedFunctions(b1,b2)
for b6, b7, b8 being Function-like quasi_total bounded Relation of b1,the carrier of b2
      st b6 = b3 & b7 = b4 & b8 = b5
   holds    b5 = b3 - b4
   iff
      for b9 being Element of b1 holds
         b8 . b9 = (b6 . b9) - (b7 . b9);

:: CSSPACE4:th 29
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
   st b2 is complete
for b3 being Function-like quasi_total Relation of NAT,the carrier of C_NormSpace_of_BoundedFunctions(b1,b2)
      st b3 is CCauchy(C_NormSpace_of_BoundedFunctions(b1,b2))
   holds b3 is convergent(C_NormSpace_of_BoundedFunctions(b1,b2));

:: CSSPACE4:th 30
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like complete CNORMSTR holds
   C_NormSpace_of_BoundedFunctions(b1,b2) is non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like complete CNORMSTR;

:: CSSPACE4:funcreg 10
registration
  let a1 be non empty set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like complete CNORMSTR;
  cluster C_NormSpace_of_BoundedFunctions(a1,a2) -> non empty complete;
end;

:: CSSPACE4:th 31
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is bounded & b2 is bounded
   holds b1 + b2 is bounded;

:: CSSPACE4:th 32
theorem
for b1 being Element of COMPLEX
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
      st b2 is bounded
   holds b1 (#) b2 is bounded;

:: CSSPACE4:th 33
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
      b1 is bounded
   iff
      |.b1.| is bounded;

:: CSSPACE4:th 34
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,COMPLEX holds
   b1 = b2 - b3
iff
   for b4 being Element of NAT holds
      b1 . b4 = (b2 . b4) - (b3 . b4);