:: Introduction to Banach and Hilbert spaces - Part II
:: by Jan Popio{\l}ek
::
:: Received July 19, 1991
:: Copyright (c) 1991-2021 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, BHSP_1, PRE_TOPC, REAL_1, NAT_1, SUBSET_1, SUPINF_2,
SEQ_2, XXREAL_0, CARD_1, METRIC_1, FUNCT_1, ARYTM_3, ARYTM_1, RELAT_1,
COMPLEX1, NORMSP_1, ORDINAL2, SEQ_1, TARSKI, STRUCT_0, XBOOLE_0;
notations TARSKI, ORDINAL1, SUBSET_1, NUMBERS, XCMPLX_0, XREAL_0, COMPLEX1,
REAL_1, NAT_1, SEQ_1, COMSEQ_2, SEQ_2, RELAT_1, VALUED_0, STRUCT_0,
PRE_TOPC, RLVECT_1, VFUNCT_1, BHSP_1, NORMSP_1, XXREAL_0;
constructors DOMAIN_1, XXREAL_0, REAL_1, NAT_1, COMPLEX1, SEQ_2, BHSP_1,
VALUED_1, RELSET_1, VFUNCT_1, COMSEQ_2;
registrations ORDINAL1, RELSET_1, XREAL_0, STRUCT_0, VFUNCT_1, FUNCT_2, NAT_1;
requirements REAL, NUMERALS, SUBSET, ARITHM;
begin
reserve X for RealUnitarySpace;
reserve x, y, z, g, g1, g2 for Point of X;
reserve a, q, r for Real;
reserve seq, seq1, seq2, seq9 for sequence of X;
reserve k, n, m, m1, m2 for Nat;
definition
let X, seq;
attr seq is convergent means
:: BHSP_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;
registration let X;
cluster constant -> convergent for sequence of X;
end;
theorem :: BHSP_2:1
seq is constant implies seq is convergent;
theorem :: BHSP_2:2
seq is convergent & (ex k st for n st k <= n holds seq9.n = seq.n
) implies seq9 is convergent;
theorem :: BHSP_2:3
seq1 is convergent & seq2 is convergent implies seq1 + seq2 is convergent;
theorem :: BHSP_2:4
seq1 is convergent & seq2 is convergent implies seq1 - seq2 is convergent;
theorem :: BHSP_2:5
seq is convergent implies a * seq is convergent;
theorem :: BHSP_2:6
seq is convergent implies - seq is convergent;
theorem :: BHSP_2:7
seq is convergent implies seq + x is convergent;
theorem :: BHSP_2:8
seq is convergent implies seq - x is convergent;
theorem :: BHSP_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
:: BHSP_2:def 2
for r st r > 0 ex m st for n st n >= m holds dist((seq.n) , it) < r;
end;
theorem :: BHSP_2:10
seq is constant & x in rng seq implies lim seq = x;
theorem :: BHSP_2:11
seq is constant & (ex n st seq.n = x) implies lim seq = x;
theorem :: BHSP_2:12
seq is convergent & (ex k st for n st n >= k holds seq9.n = seq.n)
implies lim seq = lim seq9;
theorem :: BHSP_2:13
seq1 is convergent & seq2 is convergent implies lim (seq1 + seq2
) = (lim seq1) + (lim seq2);
theorem :: BHSP_2:14
seq1 is convergent & seq2 is convergent implies lim (seq1 - seq2
) = (lim seq1) - (lim seq2);
theorem :: BHSP_2:15
seq is convergent implies lim (a * seq) = a * (lim seq);
theorem :: BHSP_2:16
seq is convergent implies lim (- seq) = - (lim seq);
theorem :: BHSP_2:17
seq is convergent implies lim (seq + x) = (lim seq) + x;
theorem :: BHSP_2:18
seq is convergent implies lim (seq - x) = (lim seq) - x;
theorem :: BHSP_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
:: BHSP_2:def 3
for n holds it.n =||.(seq.n).||;
end;
theorem :: BHSP_2:20
seq is convergent implies ||.seq.|| is convergent;
theorem :: BHSP_2:21
seq is convergent & lim seq = g implies ||.seq.|| is convergent
& lim ||.seq.|| = ||.g.||;
theorem :: BHSP_2:22
seq is convergent & lim seq = g implies ||.seq - g.|| is
convergent & lim ||.seq - g.|| = 0;
definition
let X;
let seq;
let x;
func dist(seq, x) -> Real_Sequence means
:: BHSP_2:def 4
for n holds it.n =dist((seq. n) , x);
end;
theorem :: BHSP_2:23
seq is convergent & lim seq = g implies dist(seq, g) is convergent;
theorem :: BHSP_2:24
seq is convergent & lim seq = g implies dist(seq, g) is
convergent & lim dist(seq, g) = 0;
theorem :: BHSP_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 :: BHSP_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 :: BHSP_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 :: BHSP_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 :: BHSP_2:29
seq is convergent & lim seq = g implies ||.a * seq.|| is convergent &
lim ||.a * seq.|| = ||.a * g.||;
theorem :: BHSP_2:30
seq is convergent & lim seq = g implies ||.(a * seq) - (a * g).|| is
convergent & lim ||.(a * seq) - (a * g).|| = 0;
theorem :: BHSP_2:31
seq is convergent & lim seq = g implies ||.- seq.|| is convergent &
lim ||.- seq.|| = ||.- g.||;
theorem :: BHSP_2:32
seq is convergent & lim seq = g implies ||.(- seq) - (- g).|| is
convergent & lim ||.(- seq) - (- g).|| = 0;
theorem :: BHSP_2:33
seq is convergent & lim seq = g implies ||.(seq + x) - (g + x)
.|| is convergent & lim ||.(seq + x) - (g + x).|| = 0;
theorem :: BHSP_2:34
seq is convergent & lim seq = g implies ||.seq - x.|| is convergent &
lim ||.seq - x.|| = ||.g - x.||;
theorem :: BHSP_2:35
seq is convergent & lim seq = g implies ||.(seq - x) - (g - x).|| is
convergent & lim ||.(seq - x) - (g - x).|| = 0;
theorem :: BHSP_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 :: BHSP_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 :: BHSP_2:38
seq is convergent & lim seq = g implies dist((a * seq) , (a * g)) is
convergent & lim dist((a * seq) , (a * g)) = 0;
theorem :: BHSP_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
:: BHSP_2:def 5
{y where y is Point of X : ||.x - y.||
< r};
func cl_Ball(x,r) -> Subset of X equals
:: BHSP_2:def 6
{y where y is Point of X : ||.x - y
.|| <= r};
func Sphere(x,r) -> Subset of X equals
:: BHSP_2:def 7
{y where y is Point of X : ||.x - y
.|| = r};
end;
theorem :: BHSP_2:40
z in Ball(x,r) iff ||.x - z.|| < r;
theorem :: BHSP_2:41
z in Ball(x,r) iff dist(x,z) < r;
theorem :: BHSP_2:42
r > 0 implies x in Ball(x,r);
theorem :: BHSP_2:43
y in Ball(x,r) & z in Ball(x,r) implies dist(y,z) < 2 * r;
theorem :: BHSP_2:44
y in Ball(x,r) implies y - z in Ball(x - z,r);
theorem :: BHSP_2:45
y in Ball(x,r) implies (y - x) in Ball (0.(X),r);
theorem :: BHSP_2:46
y in Ball(x,r) & r <= q implies y in Ball(x,q);
theorem :: BHSP_2:47
z in cl_Ball(x,r) iff ||.x - z.|| <= r;
theorem :: BHSP_2:48
z in cl_Ball(x,r) iff dist(x,z) <= r;
theorem :: BHSP_2:49
r >= 0 implies x in cl_Ball(x,r);
theorem :: BHSP_2:50
y in Ball(x,r) implies y in cl_Ball(x,r);
theorem :: BHSP_2:51
z in Sphere(x,r) iff ||.x - z.|| = r;
theorem :: BHSP_2:52
z in Sphere(x,r) iff dist(x,z) = r;
theorem :: BHSP_2:53
y in Sphere(x,r) implies y in cl_Ball(x,r);
theorem :: BHSP_2:54
Ball(x,r) c= cl_Ball(x,r);
theorem :: BHSP_2:55
Sphere(x,r) c= cl_Ball(x,r);
theorem :: BHSP_2:56
Ball(x,r) \/ Sphere(x,r) = cl_Ball(x,r);