Article CSSPACE2, MML version 4.99.1005

:: CSSPACE2:th 1
theorem
the carrier of Complex_l2_Space = the_set_of_l2ComplexSequences &
 (for b1 being set holds
       b1 is Element of the carrier of Complex_l2_Space
    iff
       b1 is Function-like quasi_total Relation of NAT,COMPLEX &
        |.seq_id b1.| (#) |.seq_id b1.| is summable) &
 (for b1 being set holds
       b1 is Element of the carrier of Complex_l2_Space
    iff
       b1 is Function-like quasi_total Relation of NAT,COMPLEX &
        (seq_id b1) (#) ((seq_id b1) *') is absolutely_summable) &
 0. Complex_l2_Space = CZeroseq &
 (for b1 being Element of the carrier of Complex_l2_Space holds
    b1 = seq_id b1) &
 (for b1, b2 being Element of the carrier of Complex_l2_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_l2_Space holds
    b1 * b2 = b1 (#) seq_id b2) &
 (for b1 being Element of the carrier of Complex_l2_Space holds
    - b1 = - seq_id b1 & seq_id - b1 = - seq_id b1) &
 (for b1, b2 being Element of the carrier of Complex_l2_Space holds
 b1 - b2 = (seq_id b1) - seq_id b2) &
 (for b1, b2 being Element of the carrier of Complex_l2_Space holds
 |.seq_id b1.| (#) |.seq_id b2.| is summable &
  (for b3, b4 being Element of the carrier of Complex_l2_Space holds
  b3 .|. b4 = Sum ((seq_id b3) (#) ((seq_id b4) *'))));

:: CSSPACE2:th 2
theorem
for b1, b2, b3 being Element of the carrier of Complex_l2_Space
for b4 being Element of COMPLEX holds
   (b1 .|. b1 = 0 implies b1 = 0. Complex_l2_Space) &
    (b1 = 0. Complex_l2_Space implies b1 .|. b1 = 0) &
    0 <= Re (b1 .|. b1) &
    Im (b1 .|. b1) = 0 &
    b1 .|. b2 = (b2 .|. b1) *' &
    (b1 + b2) .|. b3 = (b1 .|. b3) + (b2 .|. b3) &
    (b4 * b1) .|. b2 = b4 * (b1 .|. b2);

:: CSSPACE2:funcreg 1
registration
  cluster Complex_l2_Space -> non empty ComplexUnitarySpace-like;
end;

:: CSSPACE2:th 3
theorem
for b1 being Function-like quasi_total Relation of NAT,the carrier of Complex_l2_Space
      st b1 is Cauchy(Complex_l2_Space)
   holds b1 is convergent(Complex_l2_Space);

:: CSSPACE2:funcreg 2
registration
  cluster Complex_l2_Space -> non empty Hilbert;
end;

:: CSSPACE2:th 4
theorem
for b1, b2 being Element of COMPLEX
      st (Re b1) * Im b2 = (Re b2) * Im b1 &
         0 <= ((Re b1) * Re b2) + ((Im b1) * Im b2)
   holds |.b1 + b2.| = |.b1.| + |.b2.|;

:: CSSPACE2:th 5
theorem
for b1, b2 being Element of COMPLEX holds
2 * |.b1 * b2.| <= |.b1.| ^2 + (|.b2.| ^2);

:: CSSPACE2:th 6
theorem
for b1, b2 being Element of COMPLEX holds
|.b1 + b2.| * |.b1 + b2.| <= ((2 * |.b1.|) * |.b1.|) + ((2 * |.b2.|) * |.b2.|) &
 |.b1.| * |.b1.| <= ((2 * |.b1 - b2.|) * |.b1 - b2.|) + ((2 * |.b2.|) * |.b2.|);

:: CSSPACE2:th 7
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
   b1 = b1 *' *';

:: CSSPACE2:th 8
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
   Partial_Sums (b1 *') = (Partial_Sums b1) *';

:: CSSPACE2:th 9
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Element of NAT
      st for b3 being Element of NAT holds
           0 <= (Re b1) . b3 & (Im b1) . b3 = 0
   holds |.Partial_Sums b1.| . b2 = (Partial_Sums |.b1.|) . b2;

:: CSSPACE2:th 10
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is summable
   holds Sum (b1 *') = (Sum b1) *';

:: CSSPACE2:th 11
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is absolutely_summable
   holds |.Sum b1.| <= Sum |.b1.|;

:: CSSPACE2:th 12
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st b1 is summable &
         (for b2 being Element of NAT holds
            0 <= (Re b1) . b2 & (Im b1) . b2 = 0)
   holds |.Sum b1.| = Sum |.b1.|;

:: CSSPACE2:th 13
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
for b2 being Element of NAT holds
   0 <= (Re (b1 (#) (b1 *'))) . b2 &
    (Im (b1 (#) (b1 *'))) . b2 = 0;

:: CSSPACE2:th 14
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
   st b1 is absolutely_summable & Sum |.b1.| = 0
for b2 being Element of NAT holds
   b1 . b2 = 0c;

:: CSSPACE2:th 15
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
   |.b1.| = |.b1 *'.|;

:: CSSPACE2:th 16
theorem
for b1 being Element of COMPLEX
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
   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.|;

:: CSSPACE2:th 17
theorem
for b1 being Element of COMPLEX
for b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Function-like quasi_total Relation of NAT,COMPLEX
   st b3 is convergent & b2 is convergent
for b4 being Function-like quasi_total Relation of NAT,REAL
      st for b5 being Element of NAT holds
           b4 . b5 = (|.(b3 . b5) - b1.| * |.(b3 . b5) - b1.|) + (b2 . b5)
   holds b4 is convergent &
    lim b4 = (|.(lim b3) - b1.| * |.(lim b3) - b1.|) + lim b2;

:: CSSPACE2:th 18
theorem
for b1 being Element of COMPLEX
for b2 being Function-like quasi_total Relation of NAT,COMPLEX
   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.|;

:: CSSPACE2:th 19
theorem
for b1 being Element of COMPLEX
for b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Function-like quasi_total Relation of NAT,COMPLEX
   st b3 is convergent & b2 is convergent
for b4 being Function-like quasi_total Relation of NAT,REAL
      st for b5 being Element of NAT holds
           b4 . b5 = (|.(b3 . b5) - b1.| * |.(b3 . b5) - b1.|) + (b2 . b5)
   holds b4 is convergent &
    lim b4 = (|.(lim b3) - b1.| * |.(lim b3) - b1.|) + lim b2;