Article RSSPACE3, MML version 4.99.1005

:: RSSPACE3:funcnot 1 => RSSPACE3:func 1
definition
  func the_set_of_l1RealSequences -> Element of bool the carrier of Linear_Space_of_RealSequences means
    for b1 being set holds
          b1 in it
       iff
          b1 in the_set_of_RealSequences & seq_id b1 is absolutely_summable;
end;

:: RSSPACE3:def 1
theorem
for b1 being Element of bool the carrier of Linear_Space_of_RealSequences holds
      b1 = the_set_of_l1RealSequences
   iff
      for b2 being set holds
            b2 in b1
         iff
            b2 in the_set_of_RealSequences & seq_id b2 is absolutely_summable;

:: RSSPACE3:th 1
theorem
for b1 being Element of REAL
for b2 being Function-like quasi_total Relation of NAT,REAL
   st b2 is convergent
for b3 being Function-like quasi_total Relation of NAT,REAL
      st for b4 being Element of NAT holds
           b3 . b4 = abs ((b2 . b4) - b1)
   holds b3 is convergent & lim b3 = abs ((lim b2) - b1);

:: RSSPACE3:funcreg 1
registration
  cluster the_set_of_l1RealSequences -> non empty;
end;

:: RSSPACE3:funcreg 2
registration
  cluster the_set_of_l1RealSequences -> linearly-closed;
end;

:: RSSPACE3:funcreg 3
registration
  cluster RLSStruct(#the_set_of_l1RealSequences,Zero_(the_set_of_l1RealSequences,Linear_Space_of_RealSequences),Add_(the_set_of_l1RealSequences,Linear_Space_of_RealSequences),Mult_(the_set_of_l1RealSequences,Linear_Space_of_RealSequences)#) -> right_complementable strict Abelian add-associative right_zeroed RealLinearSpace-like;
end;

:: RSSPACE3:funcnot 2 => RSSPACE3:func 2
definition
  func l_norm -> Function-like quasi_total Relation of the_set_of_l1RealSequences,REAL means
    for b1 being set
          st b1 in the_set_of_l1RealSequences
       holds it . b1 = Sum abs seq_id b1;
end;

:: RSSPACE3:def 2
theorem
for b1 being Function-like quasi_total Relation of the_set_of_l1RealSequences,REAL holds
      b1 = l_norm
   iff
      for b2 being set
            st b2 in the_set_of_l1RealSequences
         holds b1 . b2 = Sum abs seq_id b2;

:: RSSPACE3:funcreg 4
registration
  let a1 be non empty set;
  let a2 be Element of a1;
  let 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 be Function-like quasi_total Relation of a1,REAL;
  cluster NORMSTR(#a1,a2,a3,a4,a5#) -> non empty strict;
end;

:: RSSPACE3:th 4
theorem
for b1 being NORMSTR
      st RLSStruct(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1#) is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
   holds b1 is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;

:: RSSPACE3:th 5
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st for b2 being Element of NAT holds
           b1 . b2 = 0
   holds b1 is absolutely_summable & Sum abs b1 = 0;

:: RSSPACE3:th 6
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st b1 is absolutely_summable & Sum abs b1 = 0
for b2 being Element of NAT holds
   b1 . b2 = 0;

:: RSSPACE3:th 7
theorem
NORMSTR(#the_set_of_l1RealSequences,Zero_(the_set_of_l1RealSequences,Linear_Space_of_RealSequences),Add_(the_set_of_l1RealSequences,Linear_Space_of_RealSequences),Mult_(the_set_of_l1RealSequences,Linear_Space_of_RealSequences),l_norm#) is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;

:: RSSPACE3:funcnot 3 => RSSPACE3:func 3
definition
  func l1_Space -> non empty NORMSTR equals
    NORMSTR(#the_set_of_l1RealSequences,Zero_(the_set_of_l1RealSequences,Linear_Space_of_RealSequences),Add_(the_set_of_l1RealSequences,Linear_Space_of_RealSequences),Mult_(the_set_of_l1RealSequences,Linear_Space_of_RealSequences),l_norm#);
end;

:: RSSPACE3:def 3
theorem
l1_Space = NORMSTR(#the_set_of_l1RealSequences,Zero_(the_set_of_l1RealSequences,Linear_Space_of_RealSequences),Add_(the_set_of_l1RealSequences,Linear_Space_of_RealSequences),Mult_(the_set_of_l1RealSequences,Linear_Space_of_RealSequences),l_norm#);

:: RSSPACE3:th 8
theorem
the carrier of l1_Space = the_set_of_l1RealSequences &
 (for b1 being set holds
       b1 is Element of the carrier of l1_Space
    iff
       b1 is Function-like quasi_total Relation of NAT,REAL & seq_id b1 is absolutely_summable) &
 0. l1_Space = Zeroseq &
 (for b1 being Element of the carrier of l1_Space holds
    b1 = seq_id b1) &
 (for b1, b2 being Element of the carrier of l1_Space holds
 b1 + b2 = (seq_id b1) + seq_id b2) &
 (for b1 being Element of REAL
 for b2 being Element of the carrier of l1_Space holds
    b1 * b2 = b1 (#) seq_id b2) &
 (for b1 being Element of the carrier of l1_Space holds
    - b1 = - seq_id b1 & seq_id - b1 = - seq_id b1) &
 (for b1, b2 being Element of the carrier of l1_Space holds
 b1 - b2 = (seq_id b1) - seq_id b2) &
 (for b1 being Element of the carrier of l1_Space holds
    seq_id b1 is absolutely_summable) &
 (for b1 being Element of the carrier of l1_Space holds
    ||.b1.|| = Sum abs seq_id b1);

:: RSSPACE3:th 9
theorem
for b1, b2 being Element of the carrier of l1_Space
for b3 being Element of REAL holds
   (||.b1.|| = 0 implies b1 = 0. l1_Space) &
    (b1 = 0. l1_Space implies ||.b1.|| = 0) &
    0 <= ||.b1.|| &
    ||.b1 + b2.|| <= ||.b1.|| + ||.b2.|| &
    ||.b3 * b1.|| = (abs b3) * ||.b1.||;

:: RSSPACE3:funcreg 5
registration
  cluster l1_Space -> non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like;
end;

:: RSSPACE3:funcnot 4 => RSSPACE3:func 4
definition
  let a1 be non empty NORMSTR;
  let a2, a3 be Element of the carrier of a1;
  func dist(A2,A3) -> Element of REAL equals
    ||.a2 - a3.||;
end;

:: RSSPACE3:def 4
theorem
for b1 being non empty NORMSTR
for b2, b3 being Element of the carrier of b1 holds
dist(b2,b3) = ||.b2 - b3.||;

:: RSSPACE3:attrnot 1 => RSSPACE3:attr 1
definition
  let a1 be non empty NORMSTR;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  attr a2 is CCauchy means
    for b1 being Element of REAL
          st 0 < b1
       holds ex b2 being Element of NAT st
          for b3, b4 being Element of NAT
                st b2 <= b3 & b2 <= b4
             holds dist(a2 . b3,a2 . b4) < b1;
end;

:: RSSPACE3:dfs 5
definiens
  let a1 be non empty NORMSTR;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
     a2 is CCauchy
it is sufficient to prove
  thus for b1 being Element of REAL
          st 0 < b1
       holds ex b2 being Element of NAT st
          for b3, b4 being Element of NAT
                st b2 <= b3 & b2 <= b4
             holds dist(a2 . b3,a2 . b4) < b1;

:: RSSPACE3:def 5
theorem
for b1 being non empty NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b2 is CCauchy(b1)
   iff
      for b3 being Element of REAL
            st 0 < b3
         holds ex b4 being Element of NAT st
            for b5, b6 being Element of NAT
                  st b4 <= b5 & b4 <= b6
               holds dist(b2 . b5,b2 . b6) < b3;

:: RSSPACE3:attrnot 2 => RSSPACE3:attr 1
notation
  let a1 be non empty NORMSTR;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  synonym Cauchy_sequence_by_Norm for CCauchy;
end;

:: RSSPACE3:th 10
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 holds
      b2 is CCauchy(b1)
   iff
      for b3 being Element of REAL
            st 0 < b3
         holds ex b4 being Element of NAT st
            for b5, b6 being Element of NAT
                  st b4 <= b5 & b4 <= b6
               holds ||.(b2 . b5) - (b2 . b6).|| < b3;

:: RSSPACE3:th 11
theorem
for b1 being Function-like quasi_total Relation of NAT,the carrier of l1_Space
      st b1 is CCauchy(l1_Space)
   holds b1 is convergent(l1_Space);