:: Hilbert Space of Complex Sequences
:: by Noboru Endou
::
:: Received February 24, 2004
:: Copyright (c) 2004-2021 Association of Mizar Users

Lm1: for seq being Complex_Sequence holds Partial_Sums (seq *') = (Partial_Sums seq) *'

proof end;

Lm2: for a, b being Real holds 0 <= (a ^2) + (b ^2)

proof end;

Lm3: 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;

Lm4: for seq being Complex_Sequence
for n being Nat st ( for i being Nat holds
( (Re seq) . i >= 0 & (Im seq) . i = 0 ) ) holds
|.(Partial_Sums seq).| . n = (Partial_Sums |.seq.|) . n

proof end;

Lm5: for seq being Complex_Sequence st seq is summable holds
Sum (seq *') = (Sum seq) *'

proof end;

Lm6: for seq being Complex_Sequence st seq is absolutely_summable holds
|.(Sum seq).| <= Sum |.seq.|

proof end;

Lm7: for seq being Complex_Sequence st seq is summable & ( for n being Nat holds
( (Re seq) . n >= 0 & (Im seq) . n = 0 ) ) holds
|.(Sum seq).| = Sum |.seq.|

proof end;

Lm8: for seq being Complex_Sequence
for n being Nat holds
( (Re (seq (#) (seq *'))) . n >= 0 & (Im (seq (#) (seq *'))) . n = 0 )

proof end;

Lm9: 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;

Lm10: 0. Complex_l2_Space = CZeroseq

proof end;

Lm12: for u, v being VECTOR of Complex_l2_Space holds u + v = (seq_id u) + (seq_id v)

proof end;

Lm13: for r being Complex
for u being VECTOR of Complex_l2_Space holds r * u = r (#) (seq_id u)

proof end;

Lm14: for u being VECTOR of Complex_l2_Space holds
( - u = - (seq_id u) & seq_id (- u) = - (seq_id u) )

proof end;

Lm15: for u, v being VECTOR of Complex_l2_Space holds u - v = (seq_id u) - (seq_id v)

proof end;

Lm16: for v, w being VECTOR of Complex_l2_Space holds |.(seq_id v).| (#) |.(seq_id w).| is summable

proof end;

Lm18: for seq being Complex_Sequence holds |.seq.| = |.(seq *').|

proof end;

Lm19: 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 :: 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 Lm9, Lm10, Lm12, Lm13, Lm14, Lm15, Lm16, Lm19, CSSPACE:def 17;

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 Th2, CSSPACE:def 13;

end;

Lm20: for x, y being Complex holds 2 * |.(x * y).| <= (|.x.| ^2) + (|.y.| ^2)

proof end;

Lm21: 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;

Lm22: for c being Complex
for seq being Complex_Sequence st seq is convergent holds
for rseq being Real_Sequence st ( for m being Nat holds rseq . m = |.((seq . m) - c).| * |.((seq . m) - c).| ) holds
( rseq is convergent & lim rseq = |.((lim seq) - c).| * |.((lim seq) - c).| )

proof end;

Lm23: 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 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;

registration

cluster Complex_l2_Space -> complete ;

coherence
Complex_l2_Space is complete

proof end;

end;

registration

cluster V86() right_complementable V137() V138() V139() vector-distributive scalar-distributive scalar-associative scalar-unital ComplexUnitarySpace-like complete for CUNITSTR ;

existence
ex b₁ being ComplexUnitarySpace st b₁ is complete

proof end;

end;

definition

mode ComplexHilbertSpace is complete ComplexUnitarySpace;

end;

theorem :: CSSPACE2:3

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 Lm3;

theorem :: CSSPACE2:4

for x, y being Complex holds 2 * |.(x * y).| <= (|.x.| ^2) + (|.y.| ^2) by Lm20;

theorem :: CSSPACE2:5

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 Lm21;

theorem :: CSSPACE2:6

canceled;

::$CT

theorem :: CSSPACE2:7

for seq being Complex_Sequence holds Partial_Sums (seq *') = (Partial_Sums seq) *' by Lm1;

theorem :: CSSPACE2:8

for seq being Complex_Sequence
for n being Nat st ( for i being Nat holds
( (Re seq) . i >= 0 & (Im seq) . i = 0 ) ) holds
|.(Partial_Sums seq).| . n = (Partial_Sums |.seq.|) . n by Lm4;

theorem :: CSSPACE2:9

for seq being Complex_Sequence st seq is summable holds
Sum (seq *') = (Sum seq) *' by Lm5;

theorem :: CSSPACE2:10

for seq being Complex_Sequence st seq is absolutely_summable holds
|.(Sum seq).| <= Sum |.seq.| by Lm6;

theorem :: CSSPACE2:11

for seq being Complex_Sequence st seq is summable & ( for n being Nat holds
( (Re seq) . n >= 0 & (Im seq) . n = 0 ) ) holds
|.(Sum seq).| = Sum |.seq.| by Lm7;

theorem :: CSSPACE2:12

for seq being Complex_Sequence
for n being Nat holds
( (Re (seq (#) (seq *'))) . n >= 0 & (Im (seq (#) (seq *'))) . n = 0 ) by Lm8;

theorem :: CSSPACE2:13

for seq being Complex_Sequence st seq is absolutely_summable & Sum |.seq.| = 0 holds
for n being Nat holds seq . n = 0c

proof end;

theorem :: CSSPACE2:14

for seq being Complex_Sequence holds |.seq.| = |.(seq *').| by Lm18;

theorem :: CSSPACE2:15

for c being Complex
for seq being Complex_Sequence st seq is convergent holds
for rseq being Real_Sequence st ( for m being Nat holds rseq . m = |.((seq . m) - c).| * |.((seq . m) - c).| ) holds
( rseq is convergent & lim rseq = |.((lim seq) - c).| * |.((lim seq) - c).| ) by Lm22;

theorem :: CSSPACE2:16

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 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 Lm23;