:: CLVECT_3 semantic presentation

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

scheme :: CLVECT_3:sch 1
RecFuncExCUS{ F₁() -> ComplexUnitarySpace, F₂() -> Point of F₁(), F₃( Nat, Point of F₁()) -> Point of F₁() } :

ex f being Function of NAT ,the carrier of F₁() st
( f . 0 = F₂() & ( for n being Element of NAT
for x being Point of F₁() st x = f . n holds
f . (n + 1) = F₃(n,x) ) )

proof end;

definition

let X be ComplexUnitarySpace;
let seq be sequence of X;
func Partial_Sums seq -> sequence of X means :Def1: :: CLVECT_3:def 1
( it . 0 = seq . 0 & ( for n being Nat holds it . (n + 1) = (it . n) + (seq . (n + 1)) ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def1 defines Partial_Sums CLVECT_3:def 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 :Def2: :: CLVECT_3:def 2
Partial_Sums seq is convergent;
func Sum seq -> Point of X equals :: CLVECT_3:def 3
lim (Partial_Sums seq);
correctness ;

end;

:: deftheorem Def2 defines summable CLVECT_3:def 2 :

:: deftheorem defines Sum CLVECT_3:def 3 :

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 ComplexUnitarySpace
for seq being sequence of X st X is Hilbert 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 )

proof end;

theorem :: CLVECT_3:11

for X being ComplexUnitarySpace
for seq being sequence of X st seq is summable holds
Partial_Sums seq is bounded

proof end;

theorem Th12: :: CLVECT_3:12

for X being ComplexUnitarySpace
for seq1, seq 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 4
(Partial_Sums seq) . n;
correctness ;

end;

:: deftheorem defines Sum CLVECT_3:def 4 :

theorem Th15: :: CLVECT_3:15

for X being ComplexUnitarySpace
for seq being sequence of X holds Sum seq,0 = seq . 0 by Def1;

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 Th18: :: 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 Def1;

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 5
(Sum seq,n) - (Sum seq,m);
correctness ;

end;

:: deftheorem defines Sum CLVECT_3:def 5 :

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

proof end;

theorem :: CLVECT_3:24

proof end;

definition

let Cseq be Complex_Sequence;
let n be Nat;
func Sum Cseq,n -> Complex equals :: CLVECT_3:def 6
(Partial_Sums Cseq) . n;
correctness ;

end;

:: deftheorem defines Sum CLVECT_3:def 6 :

definition

let Cseq be Complex_Sequence;
let n, m be Nat;
func Sum Cseq,n,m -> Complex equals :: CLVECT_3:def 7
(Sum Cseq,n) - (Sum Cseq,m);
correctness ;

end;

:: deftheorem defines Sum CLVECT_3:def 7 :

definition

let X be ComplexUnitarySpace;
let seq be sequence of X;
attr seq is absolutely_summable means :Def8: :: CLVECT_3:def 8
||.seq.|| is summable;

end;

:: deftheorem Def8 defines absolutely_summable CLVECT_3:def 8 :

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)).|| <= abs (((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)).|| <= abs ((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).|| <= abs (Sum ||.seq.||,m,n)

proof end;

theorem :: CLVECT_3:42

for X being ComplexUnitarySpace
for seq being sequence of X st X is Hilbert & 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 :Def9: :: CLVECT_3:def 9
for n being Nat holds it . n = (Cseq . n) * (seq . n);
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def9 defines * CLVECT_3:def 9 :

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;

Lm1: for s being Complex_Sequence st s is convergent holds
s is bounded

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 :Def10: :: CLVECT_3:def 10
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 Def10 defines Cauchy CLVECT_3:def 10 :

notation

let Cseq be Complex_Sequence;
synonym Cseq is_Cauchy_sequence for Cauchy Cseq;

end;

theorem :: CLVECT_3:51

for X being ComplexUnitarySpace
for seq being sequence of X
for Cseq being Complex_Sequence st X is Hilbert & 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;