:: CSSPACE2 semantic presentation

Lemma27: for seq being Complex_Sequence holds seq = (seq *' ) *'
proof end;

Lemma30: for seq being Complex_Sequence holds Partial_Sums (seq *' ) = (Partial_Sums seq) *'
proof end;

Lemma36: for a, b being Real holds 0 <= (a ^2 ) + (b ^2 )
proof end;

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.|
proof end;

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
proof end;

Lemma66: for seq being Complex_Sequence st seq is summable holds
Sum (seq *' ) = (Sum seq) *'
proof end;

Lemma67: for seq being Complex_Sequence st seq is absolutely_summable holds
|.(Sum seq).| <= Sum |.seq.|
proof end;

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.|
proof end;

Lemma70: for seq being Complex_Sequence
for n being Element of NAT holds
( (Re (seq (#) (seq *' ))) . n >= 0 & (Im (seq (#) (seq *' ))) . n = 0 )
proof end;

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 ) )
proof end;

Lemma72: 0. Complex_l2_Space = CZeroseq
proof end;

Lemma74: for u being VECTOR of Complex_l2_Space holds u = seq_id u
proof end;

Lemma76: for u, v being VECTOR of Complex_l2_Space holds u + v = (seq_id u) + (seq_id v)
proof end;

Lemma81: for r being Complex
for u being VECTOR of Complex_l2_Space holds r * u = r (#) (seq_id u)
proof end;

Lemma83: for u being VECTOR of Complex_l2_Space holds
( - u = - (seq_id u) & seq_id (- u) = - (seq_id u) )
proof end;

Lemma84: for u, v being VECTOR of Complex_l2_Space holds u - v = (seq_id u) - (seq_id v)
proof end;

Lemma85: for v, w being VECTOR of Complex_l2_Space holds |.(seq_id v).| (#) |.(seq_id w).| is summable
proof end;

Lemma97: for v, w being VECTOR of Complex_l2_Space holds v .|. w = Sum ((seq_id v) (#) ((seq_id w) *' ))
proof end;

Lemma98: for seq being Complex_Sequence holds |.seq.| = |.(seq *' ).|
proof end;

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 ) )
proof end;

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
for x, y, z being Point of Complex_l2_Space
for a being Complex holds
( ( x .|. x = 0 implies x = 0. Complex_l2_Space ) & ( x = 0. Complex_l2_Space implies x .|. x = 0 ) & Re (x .|. x) >= 0 & Im (x .|. x) = 0 & x .|. y = (y .|. x) *' & (x + y) .|. z = (x .|. z) + (y .|. z) & (a * x) .|. y = a * (x .|. y) )
proof end;

registration
cluster Complex_l2_Space -> ComplexUnitarySpace-like ;
coherence
Complex_l2_Space is ComplexUnitarySpace-like
by , CSSPACE:def 13;
end;

Lemma117: for x, y being Complex holds 2 * |.(x * y).| <= (|.x.| ^2 ) + (|.y.| ^2 )
proof end;

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.|) )
proof end;

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).| )
proof end;

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) )
proof end;

theorem Th3: :: CSSPACE2:3
for vseq being sequence of Complex_l2_Space st vseq is Cauchy holds
vseq is convergent
proof end;

then Lemma200: Complex_l2_Space is complete
by CLVECT_2:def 12;

registration
cluster Complex_l2_Space -> ComplexUnitarySpace-like Hilbert ;
coherence
Complex_l2_Space is Hilbert
by , CLVECT_2:def 13;
end;

theorem Th4: :: CSSPACE2:4
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.| by ;

theorem Th5: :: CSSPACE2:5
for x, y being Complex holds 2 * |.(x * y).| <= (|.x.| ^2 ) + (|.y.| ^2 ) by ;

theorem Th6: :: CSSPACE2:6
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.|) ) by ;

theorem Th7: :: CSSPACE2:7
for seq being Complex_Sequence holds seq = (seq *' ) *' by ;

theorem Th8: :: CSSPACE2:8
for seq being Complex_Sequence holds Partial_Sums (seq *' ) = (Partial_Sums seq) *' by ;

theorem Th9: :: CSSPACE2:9
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 by ;

theorem Th10: :: CSSPACE2:10
for seq being Complex_Sequence st seq is summable holds
Sum (seq *' ) = (Sum seq) *' by ;

theorem Th11: :: CSSPACE2:11
for seq being Complex_Sequence st seq is absolutely_summable holds
|.(Sum seq).| <= Sum |.seq.| by ;

theorem Th12: :: CSSPACE2:12
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.| by ;

theorem Th13: :: CSSPACE2:13
for seq being Complex_Sequence
for n being Element of NAT holds
( (Re (seq (#) (seq *' ))) . n >= 0 & (Im (seq (#) (seq *' ))) . n = 0 ) by ;

theorem Th14: :: CSSPACE2:14
for seq being Complex_Sequence st seq is absolutely_summable & Sum |.seq.| = 0 holds
for n being Element of NAT holds seq . n = 0c
proof end;

theorem Th15: :: CSSPACE2:15
for seq being Complex_Sequence holds |.seq.| = |.(seq *' ).| by ;

theorem Th16: :: CSSPACE2:16
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).| ) by ;

theorem Th17: :: CSSPACE2:17
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) ) by ;

theorem Th18: :: CSSPACE2:18
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).| ) by ;

theorem Th19: :: CSSPACE2:19
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) ) by ;