:: CLVECT_2 semantic presentation

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 Nat st
for n being Nat st n >= m holds
dist (seq . n),g < r;

end;

:: deftheorem Def1 defines convergent CLVECT_2:def 1 :

theorem Th1: :: CLVECT_2:1

for X being ComplexUnitarySpace
for seq being sequence of X st seq is constant 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 Nat st
for n being 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 Nat st
for n being 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 Nat st
for n being Nat st n >= m holds
dist (seq . n),it < r;
existence by A1, Def1;
uniqueness

proof end;

end;

:: deftheorem Def2 defines lim CLVECT_2:def 2 :

theorem Th10: :: CLVECT_2:10

for X being ComplexUnitarySpace
for x being Point of X
for seq being sequence of X st seq is constant & 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 constant & ex n being 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 Nat st
for n being 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 Nat st
for n being 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 Nat holds it . n = ||.(seq . n).||;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def3 defines ||. CLVECT_2:def 3 :

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 Nat holds it . n = dist (seq . n),x;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def4 defines dist CLVECT_2:def 4 :

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

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

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

proof end;

end;

:: deftheorem defines Ball CLVECT_2:def 5 :

:: deftheorem defines cl_Ball CLVECT_2:def 6 :

:: deftheorem defines Sphere CLVECT_2:def 7 :

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;

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 Nat st
for n, m being Nat st n >= k & m >= k holds
dist (seq . n),(seq . m) < r;

end;

:: deftheorem Def8 defines Cauchy CLVECT_2:def 8 :

theorem :: CLVECT_2:57

for X being ComplexUnitarySpace
for seq being sequence of X st seq is constant 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 Nat st
for n, m being 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 Nat st
for n being Nat st n >= m holds
dist (seq1 . n),(seq2 . n) < r;

end;

:: deftheorem Def9 defines is_compared_to CLVECT_2:def 9 :

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 by Th66;
symmetry 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 Nat st
for n being 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 Nat st
for n being 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 Nat holds ||.(seq . n).|| <= M ) );

end;

:: deftheorem Def10 defines bounded CLVECT_2:def 10 :

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 constant holds
seq is bounded

proof end;

theorem Th79: :: CLVECT_2:79

for X being ComplexUnitarySpace
for seq being sequence of X
for m being Nat ex M being Real st
( M > 0 & ( for n being 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;

definition

let X be ComplexUnitarySpace;
let Nseq be increasing Seq_of_Nat;
let seq be sequence of X;
:: original: *
redefine func seq * Nseq -> sequence of X;
coherence

proof end;

end;

theorem Th82: :: CLVECT_2:82

for X being ComplexUnitarySpace
for s being sequence of X
for N being increasing Seq_of_Nat
for n being Nat holds (s * N) . n = s . (N . n)

proof end;

theorem :: CLVECT_2:83

for X being ComplexUnitarySpace
for seq being sequence of X holds seq is subsequence of seq

proof end;

theorem :: CLVECT_2:84

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

proof end;

theorem Th85: :: CLVECT_2:85

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

proof end;

theorem :: CLVECT_2:86

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

proof end;

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;

definition

let X be ComplexUnitarySpace;
let seq be sequence of X;
let k be Nat;
func seq ^\ k -> sequence of X means :Def11: :: CLVECT_2:def 11
for n being Nat holds it . n = seq . (n + k);
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def11 defines ^\ CLVECT_2:def 11 :

theorem :: CLVECT_2:91

for X being ComplexUnitarySpace
for seq being sequence of X holds seq ^\ 0 = seq

proof end;

theorem :: CLVECT_2:92

for X being ComplexUnitarySpace
for seq being sequence of X
for k, m being Nat holds (seq ^\ k) ^\ m = (seq ^\ m) ^\ k

proof end;

theorem :: CLVECT_2:93

for X being ComplexUnitarySpace
for seq being sequence of X
for k, m being Nat holds (seq ^\ k) ^\ m = seq ^\ (k + m)

proof end;

theorem Th94: :: CLVECT_2:94

for X being ComplexUnitarySpace
for seq1, seq2 being sequence of X
for k being 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 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 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 Nat holds (z * seq) ^\ k = z * (seq ^\ k)

proof end;

theorem :: CLVECT_2:98

for X being ComplexUnitarySpace
for seq being sequence of X
for k being Nat
for Nseq being increasing Seq_of_Nat holds (seq * Nseq) ^\ k = seq * (Nseq ^\ k)

proof end;

theorem Th99: :: CLVECT_2:99

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

proof end;

theorem :: CLVECT_2:100

for X being ComplexUnitarySpace
for seq being sequence of X
for k being Nat st seq is convergent holds
( seq ^\ k is convergent & lim (seq ^\ k) = lim seq )

proof end;

theorem :: CLVECT_2:101

for X being ComplexUnitarySpace
for seq, seq1 being sequence of X st seq is convergent & ex k being 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 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 Nat st seq is Cauchy holds
seq ^\ k is Cauchy

proof end;

theorem :: CLVECT_2:104

for X being ComplexUnitarySpace
for seq1, seq2 being sequence of X
for k being 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 Nat st seq is bounded holds
seq ^\ k is bounded

proof end;

theorem :: CLVECT_2:106

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

proof end;

definition

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 Def12 defines complete CLVECT_2:def 12 :

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;

definition

let X be ComplexUnitarySpace;
attr X is Hilbert means :: CLVECT_2:def 13
( X is ComplexUnitarySpace & X is complete );

end;

:: deftheorem defines Hilbert CLVECT_2:def 13 :