:: Cauchy Sequence of Complex Unitary Space
:: by Yasumasa Suzuki and Noboru Endou
::
:: Received March 18, 2004
:: Copyright (c) 2004-2021 Association of Mizar Users

deffunc H₁( ComplexUnitarySpace) -> Element of the U1 of $1 = 0. $1;

theorem Th1: :: CLVECT_3:1

for X being ComplexUnitarySpace
for seq1, seq2 being sequence of X holds (Partial_Sums seq1) + (Partial_Sums seq2) = Partial_Sums (seq1 + seq2)

proof end;

theorem Th2: :: CLVECT_3:2

for X being ComplexUnitarySpace
for seq1, seq2 being sequence of X holds (Partial_Sums seq1) - (Partial_Sums seq2) = Partial_Sums (seq1 - seq2)

proof end;

theorem Th3: :: CLVECT_3:3

for X being ComplexUnitarySpace
for seq being sequence of X
for z being Complex holds Partial_Sums (z * seq) = z * (Partial_Sums seq)

proof end;

theorem :: CLVECT_3:4

for X being ComplexUnitarySpace
for seq being sequence of X holds Partial_Sums (- seq) = - (Partial_Sums seq)

proof end;

theorem :: CLVECT_3:5

for X being ComplexUnitarySpace
for seq1, seq2 being sequence of X
for z1, z2 being Complex holds (z1 * (Partial_Sums seq1)) + (z2 * (Partial_Sums seq2)) = Partial_Sums ((z1 * seq1) + (z2 * seq2))

proof end;

definition

let X be ComplexUnitarySpace;
let seq be sequence of X;

attr seq is summable means :Def1: :: CLVECT_3:def 1
Partial_Sums seq is convergent ;

func Sum seq -> Point of X equals :: CLVECT_3:def 2
lim (Partial_Sums seq);

correctness
coherence
lim (Partial_Sums seq) is Point of X;
;

end;

:: deftheorem Def1 defines summable CLVECT_3:def 1 :
for X being ComplexUnitarySpace
for seq being sequence of X holds
( seq is summable iff Partial_Sums seq is convergent );

:: deftheorem defines Sum CLVECT_3:def 2 :
for X being ComplexUnitarySpace
for seq being sequence of X holds Sum seq = lim (Partial_Sums seq);

theorem :: CLVECT_3:6

for X being ComplexUnitarySpace
for seq1, seq2 being sequence of X st seq1 is summable & seq2 is summable holds
( seq1 + seq2 is summable & Sum (seq1 + seq2) = (Sum seq1) + (Sum seq2) )

proof end;

theorem :: CLVECT_3:7

for X being ComplexUnitarySpace
for seq1, seq2 being sequence of X st seq1 is summable & seq2 is summable holds
( seq1 - seq2 is summable & Sum (seq1 - seq2) = (Sum seq1) - (Sum seq2) )

proof end;

theorem :: CLVECT_3:8

for X being ComplexUnitarySpace
for seq being sequence of X
for z being Complex st seq is summable holds
( z * seq is summable & Sum (z * seq) = z * (Sum seq) )

proof end;

theorem Th9: :: CLVECT_3:9

for X being ComplexUnitarySpace
for seq being sequence of X st seq is summable holds
( seq is convergent & lim seq = 0. X )

proof end;

theorem Th10: :: CLVECT_3:10

for X being ComplexHilbertSpace
for seq being sequence of X holds
( seq is summable iff for r being Real st r > 0 holds
ex k being Nat st
for n, m being Nat st n >= k & m >= k holds
||.(((Partial_Sums seq) . n) - ((Partial_Sums seq) . m)).|| < r ) by CLVECT_2:65, CLVECT_2:58, CLVECT_2:def 11;

theorem :: CLVECT_3:11

for X being ComplexUnitarySpace
for seq being sequence of X st seq is summable holds
Partial_Sums seq is bounded by CLVECT_2:80;

theorem Th12: :: CLVECT_3:12

for X being ComplexUnitarySpace
for seq, seq1 being sequence of X st ( for n being Nat holds seq1 . n = seq . 0 ) holds
Partial_Sums (seq ^\ 1) = ((Partial_Sums seq) ^\ 1) - seq1

proof end;

theorem Th13: :: CLVECT_3:13

for X being ComplexUnitarySpace
for seq being sequence of X st seq is summable holds
for k being Nat holds seq ^\ k is summable

proof end;

theorem :: CLVECT_3:14

for X being ComplexUnitarySpace
for seq being sequence of X st ex k being Nat st seq ^\ k is summable holds
seq is summable

proof end;

definition

let X be ComplexUnitarySpace;
let seq be sequence of X;
let n be Nat;

func Sum (seq,n) -> Point of X equals :: CLVECT_3:def 3
(Partial_Sums seq) . n;

correctness
coherence
(Partial_Sums seq) . n is Point of X;
;

end;

:: deftheorem defines Sum CLVECT_3:def 3 :
for X being ComplexUnitarySpace
for seq being sequence of X
for n being Nat holds Sum (seq,n) = (Partial_Sums seq) . n;

theorem :: CLVECT_3:15

for X being ComplexUnitarySpace
for seq being sequence of X holds Sum (seq,0) = seq . 0 by BHSP_4:def 1;

theorem Th16: :: CLVECT_3:16

for X being ComplexUnitarySpace
for seq being sequence of X holds Sum (seq,1) = (Sum (seq,0)) + (seq . 1)

proof end;

theorem Th17: :: CLVECT_3:17

for X being ComplexUnitarySpace
for seq being sequence of X holds Sum (seq,1) = (seq . 0) + (seq . 1)

proof end;

theorem :: CLVECT_3:18

for X being ComplexUnitarySpace
for seq being sequence of X
for n being Nat holds Sum (seq,(n + 1)) = (Sum (seq,n)) + (seq . (n + 1)) by BHSP_4:def 1;

theorem Th19: :: CLVECT_3:19

for X being ComplexUnitarySpace
for seq being sequence of X
for n being Nat holds seq . (n + 1) = (Sum (seq,(n + 1))) - (Sum (seq,n))

proof end;

theorem :: CLVECT_3:20

for X being ComplexUnitarySpace
for seq being sequence of X holds seq . 1 = (Sum (seq,1)) - (Sum (seq,0))

proof end;

definition

let X be ComplexUnitarySpace;
let seq be sequence of X;
let n, m be Nat;

func Sum (seq,n,m) -> Point of X equals :: CLVECT_3:def 4
(Sum (seq,n)) - (Sum (seq,m));

correctness
coherence
(Sum (seq,n)) - (Sum (seq,m)) is Point of X;
;

end;

:: deftheorem defines Sum CLVECT_3:def 4 :
for X being ComplexUnitarySpace
for seq being sequence of X
for n, m being Nat holds Sum (seq,n,m) = (Sum (seq,n)) - (Sum (seq,m));

theorem :: CLVECT_3:21

for X being ComplexUnitarySpace
for seq being sequence of X holds Sum (seq,1,0) = seq . 1

proof end;

theorem :: CLVECT_3:22

for X being ComplexUnitarySpace
for seq being sequence of X
for n being Nat holds Sum (seq,(n + 1),n) = seq . (n + 1) by Th19;

theorem Th23: :: CLVECT_3:23

for X being ComplexHilbertSpace
for seq being sequence of X holds
( seq is summable iff for r being Real st r > 0 holds
ex k being Nat st
for n, m being Nat st n >= k & m >= k holds
||.((Sum (seq,n)) - (Sum (seq,m))).|| < r )

proof end;

theorem :: CLVECT_3:24

for X being ComplexHilbertSpace
for seq being sequence of X holds
( seq is summable iff for r being Real st r > 0 holds
ex k being Nat st
for n, m being Nat st n >= k & m >= k holds
||.(Sum (seq,n,m)).|| < r )

proof end;

definition

let Cseq be Complex_Sequence;
let n be Nat;

func Sum (Cseq,n) -> Complex equals :: CLVECT_3:def 5
(Partial_Sums Cseq) . n;

correctness
coherence
(Partial_Sums Cseq) . n is Complex;
;

end;

:: deftheorem defines Sum CLVECT_3:def 5 :
for Cseq being Complex_Sequence
for n being Nat holds Sum (Cseq,n) = (Partial_Sums Cseq) . n;

definition

let Cseq be Complex_Sequence;
let n, m be Nat;

func Sum (Cseq,n,m) -> Complex equals :: CLVECT_3:def 6
(Sum (Cseq,n)) - (Sum (Cseq,m));

correctness
coherence
(Sum (Cseq,n)) - (Sum (Cseq,m)) is Complex;
;

end;

:: deftheorem defines Sum CLVECT_3:def 6 :
for Cseq being Complex_Sequence
for n, m being Nat holds Sum (Cseq,n,m) = (Sum (Cseq,n)) - (Sum (Cseq,m));

definition

let X be ComplexUnitarySpace;
let seq be sequence of X;

attr seq is absolutely_summable means :: CLVECT_3:def 7
||.seq.|| is summable ;

end;

:: deftheorem defines absolutely_summable CLVECT_3:def 7 :
for X being ComplexUnitarySpace
for seq being sequence of X holds
( seq is absolutely_summable iff ||.seq.|| is summable );

theorem :: CLVECT_3:25

for X being ComplexUnitarySpace
for seq1, seq2 being sequence of X st seq1 is absolutely_summable & seq2 is absolutely_summable holds
seq1 + seq2 is absolutely_summable

proof end;

theorem :: CLVECT_3:26

for X being ComplexUnitarySpace
for seq being sequence of X
for z being Complex st seq is absolutely_summable holds
z * seq is absolutely_summable

proof end;

theorem :: CLVECT_3:27

for X being ComplexUnitarySpace
for seq being sequence of X
for Rseq being Real_Sequence st ( for n being Nat holds ||.seq.|| . n <= Rseq . n ) & Rseq is summable holds
seq is absolutely_summable

proof end;

theorem :: CLVECT_3:28

for X being ComplexUnitarySpace
for seq being sequence of X
for Rseq being Real_Sequence st ( for n being Nat holds
( seq . n <> 0. X & Rseq . n = ||.(seq . (n + 1)).|| / ||.(seq . n).|| ) ) & Rseq is convergent & lim Rseq < 1 holds
seq is absolutely_summable

proof end;

theorem Th29: :: CLVECT_3:29

for X being ComplexUnitarySpace
for seq being sequence of X
for r being Real st r > 0 & ex m being Nat st
for n being Nat st n >= m holds
||.(seq . n).|| >= r & seq is convergent holds
lim seq <> 0. X

proof end;

theorem Th30: :: CLVECT_3:30

for X being ComplexUnitarySpace
for seq being sequence of X st ( for n being Nat holds seq . n <> 0. X ) & ex m being Nat st
for n being Nat st n >= m holds
||.(seq . (n + 1)).|| / ||.(seq . n).|| >= 1 holds
not seq is summable

proof end;

theorem :: CLVECT_3:31

for X being ComplexUnitarySpace
for seq being sequence of X
for Rseq being Real_Sequence st ( for n being Nat holds seq . n <> 0. X ) & ( for n being Nat holds Rseq . n = ||.(seq . (n + 1)).|| / ||.(seq . n).|| ) & Rseq is convergent & lim Rseq > 1 holds
not seq is summable

proof end;

theorem :: CLVECT_3:32

for X being ComplexUnitarySpace
for seq being sequence of X
for Rseq being Real_Sequence st ( for n being Nat holds Rseq . n = n -root ||.(seq . n).|| ) & Rseq is convergent & lim Rseq < 1 holds
seq is absolutely_summable

proof end;

theorem :: CLVECT_3:33

for X being ComplexUnitarySpace
for seq being sequence of X
for Rseq being Real_Sequence st ( for n being Nat holds Rseq . n = n -root (||.seq.|| . n) ) & ex m being Nat st
for n being Nat st n >= m holds
Rseq . n >= 1 holds
not seq is summable

proof end;

theorem :: CLVECT_3:34

for X being ComplexUnitarySpace
for seq being sequence of X
for Rseq being Real_Sequence st ( for n being Nat holds Rseq . n = n -root (||.seq.|| . n) ) & Rseq is convergent & lim Rseq > 1 holds
not seq is summable

proof end;

theorem Th35: :: CLVECT_3:35

for X being ComplexUnitarySpace
for seq being sequence of X holds Partial_Sums ||.seq.|| is non-decreasing

proof end;

theorem :: CLVECT_3:36

for X being ComplexUnitarySpace
for seq being sequence of X
for n being Nat holds (Partial_Sums ||.seq.||) . n >= 0

proof end;

theorem Th37: :: CLVECT_3:37

for X being ComplexUnitarySpace
for seq being sequence of X
for n being Nat holds ||.((Partial_Sums seq) . n).|| <= (Partial_Sums ||.seq.||) . n

proof end;

theorem :: CLVECT_3:38

for X being ComplexUnitarySpace
for seq being sequence of X
for n being Nat holds ||.(Sum (seq,n)).|| <= Sum (||.seq.||,n)

proof end;

theorem Th39: :: CLVECT_3:39

for X being ComplexUnitarySpace
for seq being sequence of X
for n, m being Nat holds ||.(((Partial_Sums seq) . m) - ((Partial_Sums seq) . n)).|| <= |.(((Partial_Sums ||.seq.||) . m) - ((Partial_Sums ||.seq.||) . n)).|

proof end;

theorem Th40: :: CLVECT_3:40

for X being ComplexUnitarySpace
for seq being sequence of X
for n, m being Nat holds ||.((Sum (seq,m)) - (Sum (seq,n))).|| <= |.((Sum (||.seq.||,m)) - (Sum (||.seq.||,n))).|

proof end;

theorem :: CLVECT_3:41

for X being ComplexUnitarySpace
for seq being sequence of X
for n, m being Nat holds ||.(Sum (seq,m,n)).|| <= |.(Sum (||.seq.||,m,n)).|

proof end;

theorem :: CLVECT_3:42

for X being ComplexHilbertSpace
for seq being sequence of X st seq is absolutely_summable holds
seq is summable

proof end;

definition

let X be ComplexUnitarySpace;
let seq be sequence of X;
let Cseq be Complex_Sequence;

func Cseq * seq -> sequence of X means :Def8: :: CLVECT_3:def 8
for n being Nat holds it . n = (Cseq . n) * (seq . n);

existence
ex b₁ being sequence of X st
for n being Nat holds b₁ . n = (Cseq . n) * (seq . n)

proof end;

uniqueness
for b₁, b₂ being sequence of X st ( for n being Nat holds b₁ . n = (Cseq . n) * (seq . n) ) & ( for n being Nat holds b₂ . n = (Cseq . n) * (seq . n) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def8 defines * CLVECT_3:def 8 :
for X being ComplexUnitarySpace
for seq being sequence of X
for Cseq being Complex_Sequence
for b₄ being sequence of X holds
( b₄ = Cseq * seq iff for n being Nat holds b₄ . n = (Cseq . n) * (seq . n) );

theorem :: CLVECT_3:43

for X being ComplexUnitarySpace
for seq1, seq2 being sequence of X
for Cseq being Complex_Sequence holds Cseq * (seq1 + seq2) = (Cseq * seq1) + (Cseq * seq2)

proof end;

theorem :: CLVECT_3:44

for X being ComplexUnitarySpace
for seq being sequence of X
for Cseq1, Cseq2 being Complex_Sequence holds (Cseq1 + Cseq2) * seq = (Cseq1 * seq) + (Cseq2 * seq)

proof end;

theorem :: CLVECT_3:45

for X being ComplexUnitarySpace
for seq being sequence of X
for Cseq1, Cseq2 being Complex_Sequence holds (Cseq1 (#) Cseq2) * seq = Cseq1 * (Cseq2 * seq)

proof end;

theorem Th46: :: CLVECT_3:46

for X being ComplexUnitarySpace
for seq being sequence of X
for Cseq being Complex_Sequence
for z being Complex holds (z (#) Cseq) * seq = z * (Cseq * seq)

proof end;

theorem :: CLVECT_3:47

for X being ComplexUnitarySpace
for seq being sequence of X
for Cseq being Complex_Sequence holds Cseq * (- seq) = (- Cseq) * seq

proof end;

theorem Th48: :: CLVECT_3:48

for X being ComplexUnitarySpace
for seq being sequence of X
for Cseq being Complex_Sequence st Cseq is convergent & seq is convergent holds
Cseq * seq is convergent

proof end;

theorem :: CLVECT_3:49

for X being ComplexUnitarySpace
for seq being sequence of X
for Cseq being Complex_Sequence st Cseq is bounded & seq is bounded holds
Cseq * seq is bounded

proof end;

theorem :: CLVECT_3:50

for X being ComplexUnitarySpace
for seq being sequence of X
for Cseq being Complex_Sequence st Cseq is convergent & seq is convergent holds
( Cseq * seq is convergent & lim (Cseq * seq) = (lim Cseq) * (lim seq) )

proof end;

definition

let Cseq be Complex_Sequence;

attr Cseq is Cauchy means :: CLVECT_3:def 9
for r being Real st r > 0 holds
ex k being Nat st
for n, m being Nat st n >= k & m >= k holds
|.((Cseq . n) - (Cseq . m)).| < r;

end;

:: deftheorem defines Cauchy CLVECT_3:def 9 :
for Cseq being Complex_Sequence holds
( Cseq is Cauchy iff for r being Real st r > 0 holds
ex k being Nat st
for n, m being Nat st n >= k & m >= k holds
|.((Cseq . n) - (Cseq . m)).| < r );

theorem :: CLVECT_3:51

for Cseq being Complex_Sequence
for X being ComplexHilbertSpace
for seq being sequence of X st seq is Cauchy & Cseq is Cauchy holds
Cseq * seq is Cauchy

proof end;

theorem Th52: :: CLVECT_3:52

for X being ComplexUnitarySpace
for seq being sequence of X
for Cseq being Complex_Sequence
for n being Nat holds (Partial_Sums ((Cseq - (Cseq ^\ 1)) * (Partial_Sums seq))) . n = ((Partial_Sums (Cseq * seq)) . (n + 1)) - ((Cseq * (Partial_Sums seq)) . (n + 1))

proof end;

theorem Th53: :: CLVECT_3:53

for X being ComplexUnitarySpace
for seq being sequence of X
for Cseq being Complex_Sequence
for n being Nat holds (Partial_Sums (Cseq * seq)) . (n + 1) = ((Cseq * (Partial_Sums seq)) . (n + 1)) - ((Partial_Sums (((Cseq ^\ 1) - Cseq) * (Partial_Sums seq))) . n)

proof end;

theorem :: CLVECT_3:54

for X being ComplexUnitarySpace
for seq being sequence of X
for Cseq being Complex_Sequence
for n being Nat holds Sum ((Cseq * seq),(n + 1)) = ((Cseq * (Partial_Sums seq)) . (n + 1)) - (Sum ((((Cseq ^\ 1) - Cseq) * (Partial_Sums seq)),n)) by Th53;