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;