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;