:: CSSPACE2 semantic presentation
Lemma27:
for seq being Complex_Sequence holds seq = (seq *' ) *'
Lemma30:
for seq being Complex_Sequence holds Partial_Sums (seq *' ) = (Partial_Sums seq) *'
Lemma36:
for a, b being Real holds 0 <= (a ^2 ) + (b ^2 )
Lemma39:
for z1, z2 being Complex st (Re z1) * (Im z2) = (Re z2) * (Im z1) & ((Re z1) * (Re z2)) + ((Im z1) * (Im z2)) >= 0 holds
|.(z1 + z2).| = |.z1.| + |.z2.|
Lemma51:
for seq being Complex_Sequence
for n being Element of NAT st ( for i being Element of NAT holds
( (Re seq) . i >= 0 & (Im seq) . i = 0 ) ) holds
|.(Partial_Sums seq).| . n = (Partial_Sums |.seq.|) . n
Lemma66:
for seq being Complex_Sequence st seq is summable holds
Sum (seq *' ) = (Sum seq) *'
Lemma67:
for seq being Complex_Sequence st seq is absolutely_summable holds
|.(Sum seq).| <= Sum |.seq.|
Lemma69:
for seq being Complex_Sequence st seq is summable & ( for n being Element of NAT holds
( (Re seq) . n >= 0 & (Im seq) . n = 0 ) ) holds
|.(Sum seq).| = Sum |.seq.|
Lemma70:
for seq being Complex_Sequence
for n being Element of NAT holds
( (Re (seq (#) (seq *' ))) . n >= 0 & (Im (seq (#) (seq *' ))) . n = 0 )
Lemma71:
for x being set holds
( x is Element of Complex_l2_Space iff ( x is Complex_Sequence & |.(seq_id x).| (#) |.(seq_id x).| is summable ) )
Lemma72:
0. Complex_l2_Space = CZeroseq
Lemma74:
for u being VECTOR of Complex_l2_Space holds u = seq_id u
Lemma76:
for u, v being VECTOR of Complex_l2_Space holds u + v = (seq_id u) + (seq_id v)
Lemma81:
for r being Complex
for u being VECTOR of Complex_l2_Space holds r * u = r (#) (seq_id u)
Lemma83:
for u being VECTOR of Complex_l2_Space holds
( - u = - (seq_id u) & seq_id (- u) = - (seq_id u) )
Lemma84:
for u, v being VECTOR of Complex_l2_Space holds u - v = (seq_id u) - (seq_id v)
Lemma85:
for v, w being VECTOR of Complex_l2_Space holds |.(seq_id v).| (#) |.(seq_id w).| is summable
Lemma97:
for v, w being VECTOR of Complex_l2_Space holds v .|. w = Sum ((seq_id v) (#) ((seq_id w) *' ))
Lemma98:
for seq being Complex_Sequence holds |.seq.| = |.(seq *' ).|
Lemma101:
for x being set holds
( x is Element of Complex_l2_Space iff ( x is Complex_Sequence & (seq_id x) (#) ((seq_id x) *' ) is absolutely_summable ) )
theorem Th1: :: CSSPACE2:1
( the
carrier of
Complex_l2_Space = the_set_of_l2ComplexSequences & ( for
x being
set holds
(
x is
Element of
Complex_l2_Space iff (
x is
Complex_Sequence &
|.(seq_id x).| (#) |.(seq_id x).| is
summable ) ) ) & ( for
x being
set holds
(
x is
Element of
Complex_l2_Space iff (
x is
Complex_Sequence &
(seq_id x) (#) ((seq_id x) *' ) is
absolutely_summable ) ) ) &
0. Complex_l2_Space = CZeroseq & ( for
u being
VECTOR of
Complex_l2_Space holds
u = seq_id u ) & ( for
u,
v being
VECTOR of
Complex_l2_Space holds
u + v = (seq_id u) + (seq_id v) ) & ( for
r being
Complex for
u being
VECTOR of
Complex_l2_Space holds
r * u = r (#) (seq_id u) ) & ( for
u being
VECTOR of
Complex_l2_Space holds
(
- u = - (seq_id u) &
seq_id (- u) = - (seq_id u) ) ) & ( for
u,
v being
VECTOR of
Complex_l2_Space holds
u - v = (seq_id u) - (seq_id v) ) & ( for
v,
w being
VECTOR of
Complex_l2_Space holds
(
|.(seq_id v).| (#) |.(seq_id w).| is
summable & ( for
v,
w being
VECTOR of
Complex_l2_Space holds
v .|. w = Sum ((seq_id v) (#) ((seq_id w) *' )) ) ) ) )
by , , , , , , , , , , CSSPACE:def 20;
theorem Th2: :: CSSPACE2:2
Lemma117:
for x, y being Complex holds 2 * |.(x * y).| <= (|.x.| ^2 ) + (|.y.| ^2 )
Lemma118:
for x, y being Complex holds
( |.(x + y).| * |.(x + y).| <= ((2 * |.x.|) * |.x.|) + ((2 * |.y.|) * |.y.|) & |.x.| * |.x.| <= ((2 * |.(x - y).|) * |.(x - y).|) + ((2 * |.y.|) * |.y.|) )
Lemma119:
for c being Complex
for seq being Complex_Sequence st seq is convergent holds
for rseq being Real_Sequence st ( for m being Element of NAT holds rseq . m = |.((seq . m) - c).| * |.((seq . m) - c).| ) holds
( rseq is convergent & lim rseq = |.((lim seq) - c).| * |.((lim seq) - c).| )
Lemma123:
for c being Complex
for seq1 being Real_Sequence
for seq being Complex_Sequence st seq is convergent & seq1 is convergent holds
for rseq being Real_Sequence st ( for i being Element of NAT holds rseq . i = (|.((seq . i) - c).| * |.((seq . i) - c).|) + (seq1 . i) ) holds
( rseq is convergent & lim rseq = (|.((lim seq) - c).| * |.((lim seq) - c).|) + (lim seq1) )
theorem Th3: :: CSSPACE2:3
then Lemma200:
Complex_l2_Space is complete
by CLVECT_2:def 12;
theorem Th4: :: CSSPACE2:4
theorem Th5: :: CSSPACE2:5
theorem Th6: :: CSSPACE2:6
theorem Th7: :: CSSPACE2:7
theorem Th8: :: CSSPACE2:8
theorem Th9: :: CSSPACE2:9
theorem Th10: :: CSSPACE2:10
theorem Th11: :: CSSPACE2:11
theorem Th12: :: CSSPACE2:12
theorem Th13: :: CSSPACE2:13
theorem Th14: :: CSSPACE2:14
theorem Th15: :: CSSPACE2:15
theorem Th16: :: CSSPACE2:16
theorem Th17: :: CSSPACE2:17
theorem Th18: :: CSSPACE2:18
theorem Th19: :: CSSPACE2:19