Article RSSPACE2, MML version 4.99.1005

:: RSSPACE2:th 1
theorem
the carrier of l2_Space = the_set_of_l2RealSequences &
 (for b1 being set holds
       b1 is Element of the carrier of l2_Space
    iff
       b1 is Function-like quasi_total Relation of NAT,REAL & (seq_id b1) (#) seq_id b1 is summable) &
 0. l2_Space = Zeroseq &
 (for b1 being Element of the carrier of l2_Space holds
    b1 = seq_id b1) &
 (for b1, b2 being Element of the carrier of l2_Space holds
 b1 + b2 = (seq_id b1) + seq_id b2) &
 (for b1 being Element of REAL
 for b2 being Element of the carrier of l2_Space holds
    b1 * b2 = b1 (#) seq_id b2) &
 (for b1 being Element of the carrier of l2_Space holds
    - b1 = - seq_id b1 & seq_id - b1 = - seq_id b1) &
 (for b1, b2 being Element of the carrier of l2_Space holds
 b1 - b2 = (seq_id b1) - seq_id b2) &
 (for b1, b2 being Element of the carrier of l2_Space holds
 (seq_id b1) (#) seq_id b2 is summable &
  (for b3, b4 being Element of the carrier of l2_Space holds
  b3 .|. b4 = Sum ((seq_id b3) (#) seq_id b4)));

:: RSSPACE2:th 2
theorem
for b1, b2, b3 being Element of the carrier of l2_Space
for b4 being Element of REAL holds
   (b1 .|. b1 = 0 implies b1 = 0. l2_Space) &
    (b1 = 0. l2_Space implies b1 .|. b1 = 0) &
    0 <= b1 .|. b1 &
    b1 .|. b2 = b2 .|. b1 &
    (b1 + b2) .|. b3 = (b1 .|. b3) + (b2 .|. b3) &
    (b4 * b1) .|. b2 = b4 * (b1 .|. b2);

:: RSSPACE2:funcreg 1
registration
  cluster l2_Space -> non empty RealUnitarySpace-like;
end;

:: RSSPACE2:th 3
theorem
for b1 being Function-like quasi_total Relation of NAT,the carrier of l2_Space
      st b1 is Cauchy(l2_Space)
   holds b1 is convergent(l2_Space);

:: RSSPACE2:funcreg 2
registration
  cluster l2_Space -> non empty complete Hilbert;
end;

:: RSSPACE2:th 4
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st for b2 being Element of NAT holds
           0 <= b1 . b2
   holds (for b2 being Element of NAT holds
       0 <= (Partial_Sums b1) . b2) &
    (for b2 being Element of NAT holds
       b1 . b2 <= (Partial_Sums b1) . b2) &
    (b1 is summable implies (for b2 being Element of NAT holds
        (Partial_Sums b1) . b2 <= Sum b1) &
     (for b2 being Element of NAT holds
        b1 . b2 <= Sum b1));

:: RSSPACE2:th 5
theorem
(for b1, b2 being Element of REAL holds
 (b1 + b2) * (b1 + b2) <= ((2 * b1) * b1) + ((2 * b2) * b2)) &
 (for b1, b2 being Element of REAL holds
 b1 * b1 <= ((2 * (b1 - b2)) * (b1 - b2)) + ((2 * b2) * b2));

:: RSSPACE2:th 6
theorem
for b1 being Element of REAL
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is convergent &
         (ex b3 being Element of NAT st
            for b4 being Element of NAT
                  st b3 <= b4
               holds b2 . b4 <= b1)
   holds lim b2 <= b1;

:: RSSPACE2:th 7
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 = ((b2 . b4) - b1) * ((b2 . b4) - b1)
   holds b3 is convergent &
    lim b3 = ((lim b2) - b1) * ((lim b2) - b1);

:: RSSPACE2:th 8
theorem
for b1 being Element of REAL
for b2, b3 being Function-like quasi_total Relation of NAT,REAL
   st b2 is convergent & b3 is convergent
for b4 being Function-like quasi_total Relation of NAT,REAL
      st for b5 being Element of NAT holds
           b4 . b5 = (((b2 . b5) - b1) * ((b2 . b5) - b1)) + (b3 . b5)
   holds b4 is convergent &
    lim b4 = (((lim b2) - b1) * ((lim b2) - b1)) + lim b3;