:: Convergent Sequences in Complex Unitary Space
:: by Noboru Endou
::
:: Received February 10, 2004
:: Copyright (c) 2004-2011 Association of Mizar Users

begin

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

definition

let X be ComplexUnitarySpace;
let seq be sequence of X;
attr seq is convergent means :Def1: :: CLVECT_2:def 1
ex g being Point of X st
for r being Real st r > 0 holds
ex m being Element of NAT st
for n being Element of NAT st n >= m holds
dist ((seq . n),g) < r;

end;

:: deftheorem Def1 defines convergent CLVECT_2:def 1 :
for X being ComplexUnitarySpace
for seq being sequence of X holds
( seq is convergent iff ex g being Point of X st
for r being Real st r > 0 holds
ex m being Element of NAT st
for n being Element of NAT st n >= m holds
dist ((seq . n),g) < r );

theorem Th1: :: CLVECT_2:1

for X being ComplexUnitarySpace
for seq being sequence of X st seq is V11() holds
seq is convergent

proof end;

theorem Th2: :: CLVECT_2:2

for X being ComplexUnitarySpace
for seq1, seq2 being sequence of X st seq1 is convergent & ex k being Element of NAT st
for n being Element of NAT st k <= n holds
seq2 . n = seq1 . n holds
seq2 is convergent

proof end;

theorem Th3: :: CLVECT_2:3

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

proof end;

theorem Th4: :: CLVECT_2:4

for X being ComplexUnitarySpace
for seq1, seq2 being sequence of X st seq1 is convergent & seq2 is convergent holds
seq1 - seq2 is convergent

proof end;

theorem Th5: :: CLVECT_2:5

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

proof end;

theorem Th6: :: CLVECT_2:6

for X being ComplexUnitarySpace
for seq being sequence of X st seq is convergent holds
- seq is convergent

proof end;

theorem Th7: :: CLVECT_2:7

for X being ComplexUnitarySpace
for x being Point of X
for seq being sequence of X st seq is convergent holds
seq + x is convergent

proof end;

theorem Th8: :: CLVECT_2:8

for X being ComplexUnitarySpace
for x being Point of X
for seq being sequence of X st seq is convergent holds
seq - x is convergent

proof end;

theorem Th9: :: CLVECT_2:9

for X being ComplexUnitarySpace
for seq being sequence of X holds
( seq is convergent iff ex g being Point of X st
for r being Real st r > 0 holds
ex m being Element of NAT st
for n being Element of NAT st n >= m holds
||.((seq . n) - g).|| < r )

proof end;

definition

let X be ComplexUnitarySpace;
let seq be sequence of X;
assume A1: seq is convergent ;
func lim seq -> Point of X means :Def2: :: CLVECT_2:def 2
for r being Real st r > 0 holds
ex m being Element of NAT st
for n being Element of NAT st n >= m holds
dist ((seq . n),it) < r;
existence
ex b₁ being Point of X st
for r being Real st r > 0 holds
ex m being Element of NAT st
for n being Element of NAT st n >= m holds
dist ((seq . n),b₁) < r by A1, Def1;
uniqueness
for b₁, b₂ being Point of X st ( for r being Real st r > 0 holds
ex m being Element of NAT st
for n being Element of NAT st n >= m holds
dist ((seq . n),b₁) < r ) & ( for r being Real st r > 0 holds
ex m being Element of NAT st
for n being Element of NAT st n >= m holds
dist ((seq . n),b₂) < r ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines lim CLVECT_2:def 2 :
for X being ComplexUnitarySpace
for seq being sequence of X st seq is convergent holds
for b₃ being Point of X holds
( b₃ = lim seq iff for r being Real st r > 0 holds
ex m being Element of NAT st
for n being Element of NAT st n >= m holds
dist ((seq . n),b₃) < r );

theorem Th10: :: CLVECT_2:10

for X being ComplexUnitarySpace
for x being Point of X
for seq being sequence of X st seq is V11() & x in rng seq holds
lim seq = x

proof end;

theorem :: CLVECT_2:11

for X being ComplexUnitarySpace
for x being Point of X
for seq being sequence of X st seq is V11() & ex n being Element of NAT st seq . n = x holds
lim seq = x

proof end;

theorem :: CLVECT_2:12

for X being ComplexUnitarySpace
for seq1, seq2 being sequence of X st seq1 is convergent & ex k being Element of NAT st
for n being Element of NAT st n >= k holds
seq2 . n = seq1 . n holds
lim seq1 = lim seq2

proof end;

theorem Th13: :: CLVECT_2:13

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

proof end;

theorem Th14: :: CLVECT_2:14

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

proof end;

theorem Th15: :: CLVECT_2:15

for X being ComplexUnitarySpace
for z being Complex
for seq being sequence of X st seq is convergent holds
lim (z * seq) = z * (lim seq)

proof end;

theorem Th16: :: CLVECT_2:16

for X being ComplexUnitarySpace
for seq being sequence of X st seq is convergent holds
lim (- seq) = - (lim seq)

proof end;

theorem Th17: :: CLVECT_2:17

for X being ComplexUnitarySpace
for x being Point of X
for seq being sequence of X st seq is convergent holds
lim (seq + x) = (lim seq) + x

proof end;

theorem :: CLVECT_2:18

for X being ComplexUnitarySpace
for x being Point of X
for seq being sequence of X st seq is convergent holds
lim (seq - x) = (lim seq) - x

proof end;

theorem Th19: :: CLVECT_2:19

for X being ComplexUnitarySpace
for g being Point of X
for seq being sequence of X st seq is convergent holds
( lim seq = g iff for r being Real st r > 0 holds
ex m being Element of NAT st
for n being Element of NAT st n >= m holds
||.((seq . n) - g).|| < r )

proof end;

definition

let X be ComplexUnitarySpace;
let seq be sequence of X;
func ||.seq.|| -> Real_Sequence means :Def3: :: CLVECT_2:def 3
for n being Element of NAT holds it . n = ||.(seq . n).||;
existence
ex b₁ being Real_Sequence st
for n being Element of NAT holds b₁ . n = ||.(seq . n).||

proof end;

uniqueness
for b₁, b₂ being Real_Sequence st ( for n being Element of NAT holds b₁ . n = ||.(seq . n).|| ) & ( for n being Element of NAT holds b₂ . n = ||.(seq . n).|| ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def3 defines ||. CLVECT_2:def 3 :
for X being ComplexUnitarySpace
for seq being sequence of X
for b₃ being Real_Sequence holds
( b₃ = ||.seq.|| iff for n being Element of NAT holds b₃ . n = ||.(seq . n).|| );

theorem Th20: :: CLVECT_2:20

for X being ComplexUnitarySpace
for seq being sequence of X st seq is convergent holds
||.seq.|| is convergent

proof end;

theorem Th21: :: CLVECT_2:21

for X being ComplexUnitarySpace
for g being Point of X
for seq being sequence of X st seq is convergent & lim seq = g holds
( ||.seq.|| is convergent & lim ||.seq.|| = ||.g.|| )

proof end;

theorem Th22: :: CLVECT_2:22

for X being ComplexUnitarySpace
for g being Point of X
for seq being sequence of X st seq is convergent & lim seq = g holds
( ||.(seq - g).|| is convergent & lim ||.(seq - g).|| = 0 )

proof end;

definition

let X be ComplexUnitarySpace;
let seq be sequence of X;
let x be Point of X;
func dist (seq,x) -> Real_Sequence means :Def4: :: CLVECT_2:def 4
for n being Element of NAT holds it . n = dist ((seq . n),x);
existence
ex b₁ being Real_Sequence st
for n being Element of NAT holds b₁ . n = dist ((seq . n),x)

proof end;

uniqueness
for b₁, b₂ being Real_Sequence st ( for n being Element of NAT holds b₁ . n = dist ((seq . n),x) ) & ( for n being Element of NAT holds b₂ . n = dist ((seq . n),x) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines dist CLVECT_2:def 4 :
for X being ComplexUnitarySpace
for seq being sequence of X
for x being Point of X
for b₄ being Real_Sequence holds
( b₄ = dist (seq,x) iff for n being Element of NAT holds b₄ . n = dist ((seq . n),x) );

theorem Th23: :: CLVECT_2:23

for X being ComplexUnitarySpace
for g being Point of X
for seq being sequence of X st seq is convergent & lim seq = g holds
dist (seq,g) is convergent

proof end;

theorem Th24: :: CLVECT_2:24

for X being ComplexUnitarySpace
for g being Point of X
for seq being sequence of X st seq is convergent & lim seq = g holds
( dist (seq,g) is convergent & lim (dist (seq,g)) = 0 )

proof end;

theorem :: CLVECT_2:25

for X being ComplexUnitarySpace
for g1, g2 being Point of X
for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds
( ||.(seq1 + seq2).|| is convergent & lim ||.(seq1 + seq2).|| = ||.(g1 + g2).|| )

proof end;

theorem :: CLVECT_2:26

for X being ComplexUnitarySpace
for g1, g2 being Point of X
for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds
( ||.((seq1 + seq2) - (g1 + g2)).|| is convergent & lim ||.((seq1 + seq2) - (g1 + g2)).|| = 0 )

proof end;

theorem :: CLVECT_2:27

for X being ComplexUnitarySpace
for g1, g2 being Point of X
for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds
( ||.(seq1 - seq2).|| is convergent & lim ||.(seq1 - seq2).|| = ||.(g1 - g2).|| )

proof end;

theorem :: CLVECT_2:28

for X being ComplexUnitarySpace
for g1, g2 being Point of X
for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds
( ||.((seq1 - seq2) - (g1 - g2)).|| is convergent & lim ||.((seq1 - seq2) - (g1 - g2)).|| = 0 )

proof end;

theorem :: CLVECT_2:29

for X being ComplexUnitarySpace
for g being Point of X
for z being Complex
for seq being sequence of X st seq is convergent & lim seq = g holds
( ||.(z * seq).|| is convergent & lim ||.(z * seq).|| = ||.(z * g).|| )

proof end;

theorem :: CLVECT_2:30

for X being ComplexUnitarySpace
for g being Point of X
for z being Complex
for seq being sequence of X st seq is convergent & lim seq = g holds
( ||.((z * seq) - (z * g)).|| is convergent & lim ||.((z * seq) - (z * g)).|| = 0 )

proof end;

theorem :: CLVECT_2:31

for X being ComplexUnitarySpace
for g being Point of X
for seq being sequence of X st seq is convergent & lim seq = g holds
( ||.(- seq).|| is convergent & lim ||.(- seq).|| = ||.(- g).|| )

proof end;

theorem :: CLVECT_2:32

for X being ComplexUnitarySpace
for g being Point of X
for seq being sequence of X st seq is convergent & lim seq = g holds
( ||.((- seq) - (- g)).|| is convergent & lim ||.((- seq) - (- g)).|| = 0 )

proof end;

Lm1: for X being ComplexUnitarySpace
for g, x being Point of X
for seq being sequence of X st seq is convergent & lim seq = g holds
( ||.(seq + x).|| is convergent & lim ||.(seq + x).|| = ||.(g + x).|| )

proof end;

theorem Th33: :: CLVECT_2:33

for X being ComplexUnitarySpace
for g, x being Point of X
for seq being sequence of X st seq is convergent & lim seq = g holds
( ||.((seq + x) - (g + x)).|| is convergent & lim ||.((seq + x) - (g + x)).|| = 0 )

proof end;

theorem :: CLVECT_2:34

for X being ComplexUnitarySpace
for g, x being Point of X
for seq being sequence of X st seq is convergent & lim seq = g holds
( ||.(seq - x).|| is convergent & lim ||.(seq - x).|| = ||.(g - x).|| )

proof end;

theorem :: CLVECT_2:35

for X being ComplexUnitarySpace
for g, x being Point of X
for seq being sequence of X st seq is convergent & lim seq = g holds
( ||.((seq - x) - (g - x)).|| is convergent & lim ||.((seq - x) - (g - x)).|| = 0 )

proof end;

theorem :: CLVECT_2:36

for X being ComplexUnitarySpace
for g1, g2 being Point of X
for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds
( dist ((seq1 + seq2),(g1 + g2)) is convergent & lim (dist ((seq1 + seq2),(g1 + g2))) = 0 )

proof end;

theorem :: CLVECT_2:37

for X being ComplexUnitarySpace
for g1, g2 being Point of X
for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds
( dist ((seq1 - seq2),(g1 - g2)) is convergent & lim (dist ((seq1 - seq2),(g1 - g2))) = 0 )

proof end;

theorem :: CLVECT_2:38

for X being ComplexUnitarySpace
for g being Point of X
for z being Complex
for seq being sequence of X st seq is convergent & lim seq = g holds
( dist ((z * seq),(z * g)) is convergent & lim (dist ((z * seq),(z * g))) = 0 )

proof end;

theorem :: CLVECT_2:39

for X being ComplexUnitarySpace
for g, x being Point of X
for seq being sequence of X st seq is convergent & lim seq = g holds
( dist ((seq + x),(g + x)) is convergent & lim (dist ((seq + x),(g + x))) = 0 )

proof end;

definition

let X be ComplexUnitarySpace;
let x be Point of X;
let r be Real;
func Ball (x,r) -> Subset of X equals :: CLVECT_2:def 5
{ y where y is Point of X : ||.(x - y).|| < r } ;
coherence
{ y where y is Point of X : ||.(x - y).|| < r } is Subset of X

proof end;

func cl_Ball (x,r) -> Subset of X equals :: CLVECT_2:def 6
{ y where y is Point of X : ||.(x - y).|| <= r } ;
coherence
{ y where y is Point of X : ||.(x - y).|| <= r } is Subset of X

proof end;

func Sphere (x,r) -> Subset of X equals :: CLVECT_2:def 7
{ y where y is Point of X : ||.(x - y).|| = r } ;
coherence
{ y where y is Point of X : ||.(x - y).|| = r } is Subset of X

proof end;

end;

:: deftheorem defines Ball CLVECT_2:def 5 :
for X being ComplexUnitarySpace
for x being Point of X
for r being Real holds Ball (x,r) = { y where y is Point of X : ||.(x - y).|| < r } ;

:: deftheorem defines cl_Ball CLVECT_2:def 6 :
for X being ComplexUnitarySpace
for x being Point of X
for r being Real holds cl_Ball (x,r) = { y where y is Point of X : ||.(x - y).|| <= r } ;

:: deftheorem defines Sphere CLVECT_2:def 7 :
for X being ComplexUnitarySpace
for x being Point of X
for r being Real holds Sphere (x,r) = { y where y is Point of X : ||.(x - y).|| = r } ;

theorem Th40: :: CLVECT_2:40

for X being ComplexUnitarySpace
for w, x being Point of X
for r being Real holds
( w in Ball (x,r) iff ||.(x - w).|| < r )

proof end;

theorem Th41: :: CLVECT_2:41

for X being ComplexUnitarySpace
for w, x being Point of X
for r being Real holds
( w in Ball (x,r) iff dist (x,w) < r )

proof end;

theorem :: CLVECT_2:42

for X being ComplexUnitarySpace
for x being Point of X
for r being Real st r > 0 holds
x in Ball (x,r)

proof end;

theorem :: CLVECT_2:43

for X being ComplexUnitarySpace
for y, x, w being Point of X
for r being Real st y in Ball (x,r) & w in Ball (x,r) holds
dist (y,w) < 2 * r

proof end;

theorem :: CLVECT_2:44

for X being ComplexUnitarySpace
for y, x, w being Point of X
for r being Real st y in Ball (x,r) holds
y - w in Ball ((x - w),r)

proof end;

theorem :: CLVECT_2:45

for X being ComplexUnitarySpace
for y, x being Point of X
for r being Real st y in Ball (x,r) holds
y - x in Ball ((0. X),r)

proof end;

theorem :: CLVECT_2:46

for X being ComplexUnitarySpace
for y, x being Point of X
for r, q being Real st y in Ball (x,r) & r <= q holds
y in Ball (x,q)

proof end;

theorem Th47: :: CLVECT_2:47

for X being ComplexUnitarySpace
for w, x being Point of X
for r being Real holds
( w in cl_Ball (x,r) iff ||.(x - w).|| <= r )

proof end;

theorem Th48: :: CLVECT_2:48

for X being ComplexUnitarySpace
for w, x being Point of X
for r being Real holds
( w in cl_Ball (x,r) iff dist (x,w) <= r )

proof end;

theorem :: CLVECT_2:49

for X being ComplexUnitarySpace
for x being Point of X
for r being Real st r >= 0 holds
x in cl_Ball (x,r)

proof end;

theorem Th50: :: CLVECT_2:50

for X being ComplexUnitarySpace
for y, x being Point of X
for r being Real st y in Ball (x,r) holds
y in cl_Ball (x,r)

proof end;

theorem Th51: :: CLVECT_2:51

for X being ComplexUnitarySpace
for w, x being Point of X
for r being Real holds
( w in Sphere (x,r) iff ||.(x - w).|| = r )

proof end;

theorem :: CLVECT_2:52

for X being ComplexUnitarySpace
for w, x being Point of X
for r being Real holds
( w in Sphere (x,r) iff dist (x,w) = r )

proof end;

theorem :: CLVECT_2:53

for X being ComplexUnitarySpace
for y, x being Point of X
for r being Real st y in Sphere (x,r) holds
y in cl_Ball (x,r)

proof end;

theorem Th54: :: CLVECT_2:54

for X being ComplexUnitarySpace
for x being Point of X
for r being Real holds Ball (x,r) c= cl_Ball (x,r)

proof end;

theorem Th55: :: CLVECT_2:55

for X being ComplexUnitarySpace
for x being Point of X
for r being Real holds Sphere (x,r) c= cl_Ball (x,r)

proof end;

theorem :: CLVECT_2:56

for X being ComplexUnitarySpace
for x being Point of X
for r being Real holds (Ball (x,r)) \/ (Sphere (x,r)) = cl_Ball (x,r)

proof end;

begin

deffunc H₂( ComplexUnitarySpace) -> Element of the carrier of $1 = 0. $1;

definition

let X be ComplexUnitarySpace;
let seq be sequence of X;
attr seq is Cauchy means :Def8: :: CLVECT_2:def 8
for r being Real st r > 0 holds
ex k being Element of NAT st
for n, m being Element of NAT st n >= k & m >= k holds
dist ((seq . n),(seq . m)) < r;

end;

:: deftheorem Def8 defines Cauchy CLVECT_2:def 8 :
for X being ComplexUnitarySpace
for seq being sequence of X holds
( seq is Cauchy iff for r being Real st r > 0 holds
ex k being Element of NAT st
for n, m being Element of NAT st n >= k & m >= k holds
dist ((seq . n),(seq . m)) < r );

theorem :: CLVECT_2:57

for X being ComplexUnitarySpace
for seq being sequence of X st seq is V11() holds
seq is Cauchy

proof end;

theorem :: CLVECT_2:58

for X being ComplexUnitarySpace
for seq being sequence of X holds
( seq is Cauchy iff for r being Real st r > 0 holds
ex k being Element of NAT st
for n, m being Element of NAT st n >= k & m >= k holds
||.((seq . n) - (seq . m)).|| < r )

proof end;

theorem :: CLVECT_2:59

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

proof end;

theorem :: CLVECT_2:60

for X being ComplexUnitarySpace
for seq1, seq2 being sequence of X st seq1 is Cauchy & seq2 is Cauchy holds
seq1 - seq2 is Cauchy

proof end;

theorem Th61: :: CLVECT_2:61

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

proof end;

theorem :: CLVECT_2:62

for X being ComplexUnitarySpace
for seq being sequence of X st seq is Cauchy holds
- seq is Cauchy

proof end;

theorem Th63: :: CLVECT_2:63

for X being ComplexUnitarySpace
for x being Point of X
for seq being sequence of X st seq is Cauchy holds
seq + x is Cauchy

proof end;

theorem :: CLVECT_2:64

for X being ComplexUnitarySpace
for x being Point of X
for seq being sequence of X st seq is Cauchy holds
seq - x is Cauchy

proof end;

theorem :: CLVECT_2:65

for X being ComplexUnitarySpace
for seq being sequence of X st seq is convergent holds
seq is Cauchy

proof end;

definition

let X be ComplexUnitarySpace;
let seq1, seq2 be sequence of X;
pred seq1 is_compared_to seq2 means :Def9: :: CLVECT_2:def 9
for r being Real st r > 0 holds
ex m being Element of NAT st
for n being Element of NAT st n >= m holds
dist ((seq1 . n),(seq2 . n)) < r;

end;

:: deftheorem Def9 defines is_compared_to CLVECT_2:def 9 :
for X being ComplexUnitarySpace
for seq1, seq2 being sequence of X holds
( seq1 is_compared_to seq2 iff for r being Real st r > 0 holds
ex m being Element of NAT st
for n being Element of NAT st n >= m holds
dist ((seq1 . n),(seq2 . n)) < r );

theorem Th66: :: CLVECT_2:66

for X being ComplexUnitarySpace
for seq being sequence of X holds seq is_compared_to seq

proof end;

theorem Th67: :: CLVECT_2:67

for X being ComplexUnitarySpace
for seq1, seq2 being sequence of X st seq1 is_compared_to seq2 holds
seq2 is_compared_to seq1

proof end;

definition

let X be ComplexUnitarySpace;
let seq1, seq2 be sequence of X;
:: original: is_compared_to
redefine pred seq1 is_compared_to seq2;
reflexivity
for seq1 being sequence of X holds seq1 is_compared_to seq1 by Th66;
symmetry
for seq1, seq2 being sequence of X st seq1 is_compared_to seq2 holds
seq2 is_compared_to seq1 by Th67;

end;

theorem :: CLVECT_2:68

for X being ComplexUnitarySpace
for seq1, seq2, seq3 being sequence of X st seq1 is_compared_to seq2 & seq2 is_compared_to seq3 holds
seq1 is_compared_to seq3

proof end;

theorem :: CLVECT_2:69

for X being ComplexUnitarySpace
for seq1, seq2 being sequence of X holds
( seq1 is_compared_to seq2 iff for r being Real st r > 0 holds
ex m being Element of NAT st
for n being Element of NAT st n >= m holds
||.((seq1 . n) - (seq2 . n)).|| < r )

proof end;

theorem :: CLVECT_2:70

for X being ComplexUnitarySpace
for seq1, seq2 being sequence of X st ex k being Element of NAT st
for n being Element of NAT st n >= k holds
seq1 . n = seq2 . n holds
seq1 is_compared_to seq2

proof end;

theorem :: CLVECT_2:71

for X being ComplexUnitarySpace
for seq1, seq2 being sequence of X st seq1 is Cauchy & seq1 is_compared_to seq2 holds
seq2 is Cauchy

proof end;

theorem :: CLVECT_2:72

for X being ComplexUnitarySpace
for seq1, seq2 being sequence of X st seq1 is convergent & seq1 is_compared_to seq2 holds
seq2 is convergent

proof end;

theorem :: CLVECT_2:73

for X being ComplexUnitarySpace
for g being Point of X
for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g & seq1 is_compared_to seq2 holds
( seq2 is convergent & lim seq2 = g )

proof end;

definition

let X be ComplexUnitarySpace;
let seq be sequence of X;
attr seq is bounded means :Def10: :: CLVECT_2:def 10
ex M being Real st
( M > 0 & ( for n being Element of NAT holds ||.(seq . n).|| <= M ) );

end;

:: deftheorem Def10 defines bounded CLVECT_2:def 10 :
for X being ComplexUnitarySpace
for seq being sequence of X holds
( seq is bounded iff ex M being Real st
( M > 0 & ( for n being Element of NAT holds ||.(seq . n).|| <= M ) ) );

theorem Th74: :: CLVECT_2:74

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

proof end;

theorem Th75: :: CLVECT_2:75

for X being ComplexUnitarySpace
for seq being sequence of X st seq is bounded holds
- seq is bounded

proof end;

theorem :: CLVECT_2:76

for X being ComplexUnitarySpace
for seq1, seq2 being sequence of X st seq1 is bounded & seq2 is bounded holds
seq1 - seq2 is bounded

proof end;

theorem :: CLVECT_2:77

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

proof end;

theorem :: CLVECT_2:78

for X being ComplexUnitarySpace
for seq being sequence of X st seq is V11() holds
seq is bounded

proof end;

theorem Th79: :: CLVECT_2:79

for X being ComplexUnitarySpace
for seq being sequence of X
for m being Element of NAT ex M being Real st
( M > 0 & ( for n being Element of NAT st n <= m holds
||.(seq . n).|| < M ) )

proof end;

theorem Th80: :: CLVECT_2:80

for X being ComplexUnitarySpace
for seq being sequence of X st seq is convergent holds
seq is bounded

proof end;

theorem :: CLVECT_2:81

for X being ComplexUnitarySpace
for seq1, seq2 being sequence of X st seq1 is bounded & seq1 is_compared_to seq2 holds
seq2 is bounded

proof end;

theorem :: CLVECT_2:82

canceled;

theorem :: CLVECT_2:83

canceled;

theorem :: CLVECT_2:84

canceled;

theorem :: CLVECT_2:85

canceled;

theorem :: CLVECT_2:86

canceled;

theorem Th87: :: CLVECT_2:87

for X being ComplexUnitarySpace
for seq, seq1 being sequence of X st seq is bounded & seq1 is subsequence of seq holds
seq1 is bounded

proof end;

theorem Th88: :: CLVECT_2:88

for X being ComplexUnitarySpace
for seq, seq1 being sequence of X st seq is convergent & seq1 is subsequence of seq holds
seq1 is convergent

proof end;

theorem Th89: :: CLVECT_2:89

for X being ComplexUnitarySpace
for seq1, seq being sequence of X st seq1 is subsequence of seq & seq is convergent holds
lim seq1 = lim seq

proof end;

theorem Th90: :: CLVECT_2:90

for X being ComplexUnitarySpace
for seq, seq1 being sequence of X st seq is Cauchy & seq1 is subsequence of seq holds
seq1 is Cauchy

proof end;

theorem :: CLVECT_2:91

canceled;

theorem :: CLVECT_2:92

canceled;

theorem :: CLVECT_2:93

canceled;

theorem Th94: :: CLVECT_2:94

for X being ComplexUnitarySpace
for seq1, seq2 being sequence of X
for k being Element of NAT holds (seq1 + seq2) ^\ k = (seq1 ^\ k) + (seq2 ^\ k)

proof end;

theorem Th95: :: CLVECT_2:95

for X being ComplexUnitarySpace
for seq being sequence of X
for k being Element of NAT holds (- seq) ^\ k = - (seq ^\ k)

proof end;

theorem :: CLVECT_2:96

for X being ComplexUnitarySpace
for seq1, seq2 being sequence of X
for k being Element of NAT holds (seq1 - seq2) ^\ k = (seq1 ^\ k) - (seq2 ^\ k)

proof end;

theorem :: CLVECT_2:97

for X being ComplexUnitarySpace
for z being Complex
for seq being sequence of X
for k being Element of NAT holds (z * seq) ^\ k = z * (seq ^\ k)

proof end;

theorem :: CLVECT_2:98

canceled;

theorem :: CLVECT_2:99

canceled;

theorem :: CLVECT_2:100

for X being ComplexUnitarySpace
for seq being sequence of X
for k being Element of NAT st seq is convergent holds
( seq ^\ k is convergent & lim (seq ^\ k) = lim seq ) by Th88, Th89;

theorem :: CLVECT_2:101

for X being ComplexUnitarySpace
for seq, seq1 being sequence of X st seq is convergent & ex k being Element of NAT st seq = seq1 ^\ k holds
seq1 is convergent

proof end;

theorem :: CLVECT_2:102

for X being ComplexUnitarySpace
for seq, seq1 being sequence of X st seq is Cauchy & ex k being Element of NAT st seq = seq1 ^\ k holds
seq1 is Cauchy

proof end;

theorem :: CLVECT_2:103

for X being ComplexUnitarySpace
for seq being sequence of X
for k being Element of NAT st seq is Cauchy holds
seq ^\ k is Cauchy by Th90;

theorem :: CLVECT_2:104

for X being ComplexUnitarySpace
for seq1, seq2 being sequence of X
for k being Element of NAT st seq1 is_compared_to seq2 holds
seq1 ^\ k is_compared_to seq2 ^\ k

proof end;

theorem :: CLVECT_2:105

for X being ComplexUnitarySpace
for seq being sequence of X
for k being Element of NAT st seq is bounded holds
seq ^\ k is bounded by Th87;

theorem :: CLVECT_2:106

for X being ComplexUnitarySpace
for seq being sequence of X
for k being Element of NAT st seq is V11() holds
seq ^\ k is V11() ;

definition

canceled;
let X be ComplexUnitarySpace;
attr X is complete means :Def12: :: CLVECT_2:def 12
for seq being sequence of X st seq is Cauchy holds
seq is convergent ;

end;

:: deftheorem CLVECT_2:def 11 :
canceled;

:: deftheorem Def12 defines complete CLVECT_2:def 12 :
for X being ComplexUnitarySpace holds
( X is complete iff for seq being sequence of X st seq is Cauchy holds
seq is convergent );

theorem :: CLVECT_2:107

for X being ComplexUnitarySpace
for seq being sequence of X st X is complete & seq is Cauchy holds
seq is bounded

proof end;