:: Convergent Sequences in Complex Unitary Space
:: by Noboru Endou
::
:: Received February 10, 2004
:: Copyright (c) 2004-2018 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, CSSPACE, PRE_TOPC, XCMPLX_0, REAL_1, NAT_1, SUBSET_1,
ORDINAL2, SUPINF_2, SEQ_2, XXREAL_0, CARD_1, METRIC_1, FUNCT_1, ARYTM_3,
ARYTM_1, RELAT_1, COMPLEX1, NORMSP_1, SEQ_1, TARSKI, STRUCT_0, XBOOLE_0,
BHSP_3, XXREAL_2, VALUED_0, REWRITE1;
notations TARSKI, ORDINAL1, SUBSET_1, NUMBERS, XXREAL_0, XCMPLX_0, XREAL_0,
COMPLEX1, REAL_1, RELAT_1, NAT_1, SEQ_1, VALUED_0, STRUCT_0, PRE_TOPC,
RLVECT_1, VFUNCT_1, NORMSP_1, BHSP_1, SEQ_2, CLVECT_1, CSSPACE, RECDEF_1;
constructors REAL_1, COMPLEX1, SEQ_2, BHSP_3, CSSPACE, VALUED_1, RECDEF_1,
RELSET_1, VFUNCT_1, COMSEQ_2;
registrations XBOOLE_0, ORDINAL1, RELSET_1, XREAL_0, NAT_1, MEMBERED,
STRUCT_0, VALUED_0, VFUNCT_1, FUNCT_2;
requirements SUBSET, REAL, BOOLE, NUMERALS, ARITHM;
begin :: Convergence in complex unitary space
reserve X for ComplexUnitarySpace;
reserve x, y, w, g, g1, g2 for Point of X;
reserve z for Complex;
reserve p, q, r, M, M1, M2 for Real;
reserve seq, seq1, seq2, seq3 for sequence of X;
reserve k,n,m for Nat;
reserve Nseq for increasing sequence of NAT;
definition
let X, seq;
attr seq is convergent means
:: CLVECT_2:def 1
ex g st for r st r > 0 ex m st for n st n >= m holds dist(seq.n, g) < r;
end;
theorem :: CLVECT_2:1
seq is constant implies seq is convergent;
theorem :: CLVECT_2:2
seq1 is convergent & (ex k st for n st k <= n holds seq2.n = seq1
.n) implies seq2 is convergent;
theorem :: CLVECT_2:3
seq1 is convergent & seq2 is convergent implies seq1 + seq2 is convergent;
theorem :: CLVECT_2:4
seq1 is convergent & seq2 is convergent implies seq1 - seq2 is convergent;
theorem :: CLVECT_2:5
seq is convergent implies z * seq is convergent;
theorem :: CLVECT_2:6
seq is convergent implies - seq is convergent;
theorem :: CLVECT_2:7
seq is convergent implies seq + x is convergent;
theorem :: CLVECT_2:8
seq is convergent implies seq - x is convergent;
theorem :: CLVECT_2:9
seq is convergent iff ex g st for r st r > 0 ex m st for n st n
>= m holds ||.(seq.n) - g.|| < r;
definition
let X, seq;
assume
seq is convergent;
func lim seq -> Point of X means
:: CLVECT_2:def 2
for r st r > 0 ex m st for n st n >= m holds dist((seq.n) , it) < r;
end;
theorem :: CLVECT_2:10
seq is constant & x in rng seq implies lim seq = x;
theorem :: CLVECT_2:11
seq is constant & (ex n st seq.n = x) implies lim seq = x;
theorem :: CLVECT_2:12
seq1 is convergent & (ex k st for n st n >= k holds seq2.n = seq1.n)
implies lim seq1 = lim seq2;
theorem :: CLVECT_2:13
seq1 is convergent & seq2 is convergent implies lim (seq1 + seq2
) = (lim seq1) + (lim seq2);
theorem :: CLVECT_2:14
seq1 is convergent & seq2 is convergent implies lim (seq1 - seq2
) = (lim seq1) - (lim seq2);
theorem :: CLVECT_2:15
seq is convergent implies lim (z * seq) = z * (lim seq);
theorem :: CLVECT_2:16
seq is convergent implies lim (- seq) = - (lim seq);
theorem :: CLVECT_2:17
seq is convergent implies lim (seq + x) = (lim seq) + x;
theorem :: CLVECT_2:18
seq is convergent implies lim (seq - x) = (lim seq) - x;
theorem :: CLVECT_2:19
seq is convergent implies (lim seq = g iff for r st r > 0 ex m
st for n st n >= m holds ||.(seq.n) - g.|| < r);
definition
let X, seq;
func ||.seq.|| -> Real_Sequence means
:: CLVECT_2:def 3
for n holds it.n =||.(seq.n).||;
end;
theorem :: CLVECT_2:20
seq is convergent implies ||.seq.|| is convergent;
theorem :: CLVECT_2:21
seq is convergent & lim seq = g implies ||.seq.|| is convergent
& lim ||.seq.|| = ||.g.||;
theorem :: CLVECT_2:22
seq is convergent & lim seq = g implies ||.seq - g.|| is
convergent & lim ||.seq - g.|| = 0;
definition
let X, seq, x;
func dist(seq, x) -> Real_Sequence means
:: CLVECT_2:def 4
for n holds it.n =dist((seq. n) , x);
end;
theorem :: CLVECT_2:23
seq is convergent & lim seq = g implies dist(seq,g) is convergent;
theorem :: CLVECT_2:24
seq is convergent & lim seq = g implies dist(seq,g) is
convergent & lim dist(seq,g) = 0;
theorem :: CLVECT_2:25
seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 =
g2 implies ||.seq1 + seq2.|| is convergent & lim ||.seq1 + seq2.|| = ||.g1 + g2
.||;
theorem :: CLVECT_2:26
seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 =
g2 implies ||.(seq1 + seq2) - (g1 + g2).|| is convergent & lim ||.(seq1 + seq2)
- (g1 + g2).|| = 0;
theorem :: CLVECT_2:27
seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 =
g2 implies ||.seq1 - seq2.|| is convergent & lim ||.seq1 - seq2.|| = ||.g1 - g2
.||;
theorem :: CLVECT_2:28
seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 =
g2 implies ||.(seq1 - seq2) - (g1 - g2).|| is convergent & lim ||.(seq1 - seq2)
- (g1 - g2).|| = 0;
theorem :: CLVECT_2:29
seq is convergent & lim seq = g implies ||.z * seq.|| is convergent &
lim ||.z * seq.|| = ||.z * g.||;
theorem :: CLVECT_2:30
seq is convergent & lim seq = g implies ||.(z * seq) - (z * g).|| is
convergent & lim ||.(z * seq) - (z * g).|| = 0;
theorem :: CLVECT_2:31
seq is convergent & lim seq = g implies ||.- seq.|| is convergent &
lim ||.- seq.|| = ||.- g.||;
theorem :: CLVECT_2:32
seq is convergent & lim seq = g implies ||.(- seq) - (- g).|| is
convergent & lim ||.(- seq) - (- g).|| = 0;
theorem :: CLVECT_2:33
seq is convergent & lim seq = g implies ||.(seq + x) - (g + x)
.|| is convergent & lim ||.(seq + x) - (g + x).|| = 0;
theorem :: CLVECT_2:34
seq is convergent & lim seq = g implies ||.seq - x.|| is convergent &
lim ||.seq - x.|| = ||.g - x.||;
theorem :: CLVECT_2:35
seq is convergent & lim seq = g implies ||.(seq - x) - (g - x).|| is
convergent & lim ||.(seq - x) - (g - x).|| = 0;
theorem :: CLVECT_2:36
seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 =
g2 implies dist((seq1 + seq2) , (g1 + g2)) is convergent & lim dist((seq1 +
seq2) , (g1 + g2)) = 0;
theorem :: CLVECT_2:37
seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 =
g2 implies dist((seq1 - seq2) , (g1 - g2)) is convergent & lim dist((seq1 -
seq2) , (g1 - g2)) = 0;
theorem :: CLVECT_2:38
seq is convergent & lim seq = g implies dist((z * seq) , (z * g)) is
convergent & lim dist((z * seq) , (z * g)) = 0;
theorem :: CLVECT_2:39
seq is convergent & lim seq = g implies dist((seq + x) , (g + x)) is
convergent & lim dist((seq + x) , (g + x)) = 0;
definition
let X, x, r;
func Ball(x,r) -> Subset of X equals
:: CLVECT_2:def 5
{y where y is Point of X : ||.x - y.||
< r};
func cl_Ball(x,r) -> Subset of X equals
:: CLVECT_2:def 6
{y where y is Point of X : ||.x - y
.|| <= r};
func Sphere(x,r) -> Subset of X equals
:: CLVECT_2:def 7
{y where y is Point of X : ||.x - y
.|| = r};
end;
theorem :: CLVECT_2:40
w in Ball(x,r) iff ||.x - w.|| < r;
theorem :: CLVECT_2:41
w in Ball(x,r) iff dist(x,w) < r;
theorem :: CLVECT_2:42
r > 0 implies x in Ball(x,r);
theorem :: CLVECT_2:43
y in Ball(x,r) & w in Ball(x,r) implies dist(y,w) < 2 * r;
theorem :: CLVECT_2:44
y in Ball(x,r) implies y - w in Ball(x - w,r);
theorem :: CLVECT_2:45
y in Ball(x,r) implies (y - x) in Ball (0.(X),r);
theorem :: CLVECT_2:46
y in Ball(x,r) & r <= q implies y in Ball(x,q);
theorem :: CLVECT_2:47
w in cl_Ball(x,r) iff ||.x - w.|| <= r;
theorem :: CLVECT_2:48
w in cl_Ball(x,r) iff dist(x,w) <= r;
theorem :: CLVECT_2:49
r >= 0 implies x in cl_Ball(x,r);
theorem :: CLVECT_2:50
y in Ball(x,r) implies y in cl_Ball(x,r);
theorem :: CLVECT_2:51
w in Sphere(x,r) iff ||.x - w.|| = r;
theorem :: CLVECT_2:52
w in Sphere(x,r) iff dist(x,w) = r;
theorem :: CLVECT_2:53
y in Sphere(x,r) implies y in cl_Ball(x,r);
theorem :: CLVECT_2:54
Ball(x,r) c= cl_Ball(x,r);
theorem :: CLVECT_2:55
Sphere(x,r) c= cl_Ball(x,r);
theorem :: CLVECT_2:56
Ball(x,r) \/ Sphere(x,r) = cl_Ball(x,r);
begin
definition
let X;
let seq;
attr seq is Cauchy means
:: CLVECT_2:def 8
for r st r > 0 ex k st for n, m st n >= k &
m >= k holds dist((seq.n), (seq.m)) < r;
end;
theorem :: CLVECT_2:57
seq is constant implies seq is Cauchy;
theorem :: CLVECT_2:58
seq is Cauchy iff for r st r > 0 ex k st for n, m st n >= k & m >= k
holds ||.(seq.n) - (seq.m).|| < r;
theorem :: CLVECT_2:59
seq1 is Cauchy & seq2 is Cauchy implies seq1 + seq2 is Cauchy;
theorem :: CLVECT_2:60
seq1 is Cauchy & seq2 is Cauchy implies seq1 - seq2 is Cauchy;
theorem :: CLVECT_2:61
seq is Cauchy implies z * seq is Cauchy;
theorem :: CLVECT_2:62
seq is Cauchy implies - seq is Cauchy;
theorem :: CLVECT_2:63
seq is Cauchy implies seq + x is Cauchy;
theorem :: CLVECT_2:64
seq is Cauchy implies seq - x is Cauchy;
theorem :: CLVECT_2:65
seq is convergent implies seq is Cauchy;
definition
let X;
let seq1, seq2;
pred seq1 is_compared_to seq2 means
:: CLVECT_2:def 9
for r st r > 0 ex m st for n st n >= m holds dist(seq1.n, seq2.n) < r;
end;
theorem :: CLVECT_2:66
seq is_compared_to seq;
theorem :: CLVECT_2:67
seq1 is_compared_to seq2 implies seq2 is_compared_to seq1;
definition
let X;
let seq1, seq2;
redefine pred seq1 is_compared_to seq2;
reflexivity;
symmetry;
end;
theorem :: CLVECT_2:68
seq1 is_compared_to seq2 & seq2 is_compared_to seq3 implies seq1
is_compared_to seq3;
theorem :: CLVECT_2:69
seq1 is_compared_to seq2 iff for r st r > 0 ex m st for n st n >= m
holds ||.(seq1.n) - (seq2.n).|| < r;
theorem :: CLVECT_2:70
( ex k st for n st n >= k holds seq1.n = seq2.n ) implies seq1
is_compared_to seq2;
theorem :: CLVECT_2:71
seq1 is Cauchy & seq1 is_compared_to seq2 implies seq2 is Cauchy;
theorem :: CLVECT_2:72
seq1 is convergent & seq1 is_compared_to seq2 implies seq2 is convergent;
theorem :: CLVECT_2:73
seq1 is convergent & lim seq1 = g & seq1 is_compared_to seq2 implies
seq2 is convergent & lim seq2 = g;
definition
let X;
let seq;
attr seq is bounded means
:: CLVECT_2:def 10
ex M st M > 0 & for n holds ||.seq.n.|| <= M;
end;
theorem :: CLVECT_2:74
seq1 is bounded & seq2 is bounded implies seq1 + seq2 is bounded;
theorem :: CLVECT_2:75
seq is bounded implies -seq is bounded;
theorem :: CLVECT_2:76
seq1 is bounded & seq2 is bounded implies seq1 - seq2 is bounded;
theorem :: CLVECT_2:77
seq is bounded implies z * seq is bounded;
theorem :: CLVECT_2:78
seq is constant implies seq is bounded;
theorem :: CLVECT_2:79
for m ex M st ( M > 0 & for n st n <= m holds ||.seq.n.|| < M );
theorem :: CLVECT_2:80
seq is convergent implies seq is bounded;
theorem :: CLVECT_2:81
seq1 is bounded & seq1 is_compared_to seq2 implies seq2 is bounded;
theorem :: CLVECT_2:82
seq is bounded & seq1 is subsequence of seq implies seq1 is bounded;
theorem :: CLVECT_2:83
seq is convergent & seq1 is subsequence of seq implies seq1 is convergent;
theorem :: CLVECT_2:84
seq1 is subsequence of seq & seq is convergent implies lim seq1= lim seq;
theorem :: CLVECT_2:85
seq is Cauchy & seq1 is subsequence of seq implies seq1 is Cauchy;
theorem :: CLVECT_2:86
(seq1 + seq2) ^\k = (seq1 ^\k) + (seq2 ^\k);
theorem :: CLVECT_2:87
(-seq) ^\k = -(seq ^\k);
theorem :: CLVECT_2:88
(seq1 - seq2) ^\k = (seq1 ^\k) - (seq2 ^\k);
theorem :: CLVECT_2:89
(z * seq) ^\k = z * (seq ^\k);
theorem :: CLVECT_2:90
seq is convergent implies (seq ^\k) is convergent & lim (seq ^\k)=lim
seq;
theorem :: CLVECT_2:91
seq is convergent & (ex k st seq = seq1 ^\k) implies seq1 is convergent;
theorem :: CLVECT_2:92
seq is Cauchy & (ex k st seq = seq1 ^\k) implies seq1 is Cauchy;
theorem :: CLVECT_2:93
seq is Cauchy implies (seq ^\k) is Cauchy;
theorem :: CLVECT_2:94
seq1 is_compared_to seq2 implies (seq1 ^\k) is_compared_to (seq2 ^\k);
theorem :: CLVECT_2:95
seq is bounded implies (seq ^\k) is bounded;
theorem :: CLVECT_2:96
seq is constant implies (seq ^\k) is constant;
definition
let X;
attr X is complete means
:: CLVECT_2:def 11
for seq holds seq is Cauchy implies seq is convergent;
end;
theorem :: CLVECT_2:97
X is complete & seq is Cauchy implies seq is bounded;