:: Introduction to Banach and Hilbert spaces - Part III
:: 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, ORDINAL2, SUBSET_1,
SUPINF_2, XXREAL_0, CARD_1, METRIC_1, FUNCT_1, ARYTM_3, NORMSP_1,
ARYTM_1, RELAT_1, COMPLEX1, SEQ_2, XXREAL_2, VALUED_0, REWRITE1, BHSP_3;
notations ORDINAL1, SUBSET_1, NUMBERS, XCMPLX_0, XREAL_0, COMPLEX1, REAL_1,
XXREAL_0, NAT_1, VALUED_0, STRUCT_0, PRE_TOPC, RLVECT_1, VFUNCT_1,
NORMSP_1, BHSP_1, BHSP_2;
constructors PARTFUN1, FUNCT_3, XXREAL_0, REAL_1, COMPLEX1, SEQM_3, BHSP_2,
VALUED_1, RECDEF_1, NAT_1, RELSET_1, VFUNCT_1;
registrations XBOOLE_0, ORDINAL1, RELSET_1, XREAL_0, NAT_1, STRUCT_0,
VALUED_0, VFUNCT_1, FUNCT_2, BHSP_2;
requirements REAL, NUMERALS, SUBSET, ARITHM;
begin
reserve X for RealUnitarySpace,
x, g, g1, h for Point of X,
a, p, r, M, M1, M2 for Real,
seq, seq1, seq2, seq3 for sequence of X,
Nseq for increasing sequence of NAT,
k, l, l1, l2, l3, n, m, m1, m2 for Nat;
registration let X;
cluster constant for sequence of X;
end;
definition
let X;
let seq;
attr seq is Cauchy means
:: BHSP_3:def 1
for r st r > 0 ex k st for n, m st n >= k &
m >= k holds dist(seq.n, seq.m) < r;
end;
registration let X;
cluster constant -> Cauchy for sequence of X;
end;
::$CT
theorem :: BHSP_3:2
seq is Cauchy iff for r st r > 0 ex k st for n, m st n >= k & m >= k
holds ||.(seq.n) - (seq.m).|| < r;
registration let X; let seq1,seq2 be Cauchy sequence of X;
cluster seq1+seq2 -> Cauchy;
end;
registration let X; let seq1,seq2 be Cauchy sequence of X;
cluster seq1-seq2 -> Cauchy;
end;
registration let X,a; let seq be Cauchy sequence of X;
cluster a * seq -> Cauchy;
end;
registration let X; let seq be Cauchy sequence of X;
cluster - seq -> Cauchy;
end;
registration let X,x; let seq be Cauchy sequence of X;
cluster seq + x -> Cauchy;
end;
registration let X,x; let seq be Cauchy sequence of X;
cluster seq - x -> Cauchy;
end;
registration let X;
cluster convergent -> Cauchy for sequence of X;
end;
definition
let X;
let seq1, seq2;
pred seq1 is_compared_to seq2 means
:: BHSP_3:def 2
for r st r > 0 ex m st for n st n >= m holds dist(seq1.n, seq2.n) < r;
reflexivity;
symmetry;
end;
::$CT 9
theorem :: BHSP_3:12
seq1 is_compared_to seq2 & seq2 is_compared_to seq3 implies seq1
is_compared_to seq3;
theorem :: BHSP_3:13
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 :: BHSP_3:14
( ex k st for n st n >= k holds seq1.n = seq2.n ) implies seq1
is_compared_to seq2;
theorem :: BHSP_3:15
seq1 is Cauchy & seq1 is_compared_to seq2 implies seq2 is Cauchy;
theorem :: BHSP_3:16
seq1 is convergent & seq1 is_compared_to seq2 implies seq2 is convergent;
theorem :: BHSP_3:17
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
:: BHSP_3:def 3
ex M st M > 0 & for n holds ||.seq.n.|| <= M;
end;
registration let X;
cluster constant -> bounded for sequence of X;
end;
registration let X; let seq1, seq2 be bounded sequence of X;
cluster seq1 + seq2 -> bounded;
end;
registration let X; let seq be bounded sequence of X;
cluster -seq -> bounded;
end;
registration let X; let seq1, seq2 be bounded sequence of X;
cluster seq1 - seq2 -> bounded;
end;
registration let X,a; let seq be bounded sequence of X;
cluster a * seq -> bounded;
end;
::$CT 5
theorem :: BHSP_3:23
for m ex M st ( M > 0 & for n st n <= m holds ||.seq.n.|| < M );
registration let X;
cluster convergent -> bounded for sequence of X;
end;
::$CT
theorem :: BHSP_3:25
seq1 is bounded & seq1 is_compared_to seq2 implies seq2 is bounded;
registration let X; let seq be bounded sequence of X;
cluster -> bounded for subsequence of seq;
end;
registration let X; let seq be convergent sequence of X;
cluster -> convergent for subsequence of seq;
end;
::$CT 2
theorem :: BHSP_3:28
seq1 is subsequence of seq & seq is convergent implies lim seq1= lim seq;
registration let X; let seq be Cauchy sequence of X;
cluster -> Cauchy for subsequence of seq;
end;
::$CT
theorem :: BHSP_3:30
(seq ^\k)^\m = (seq ^\m)^\k;
theorem :: BHSP_3:31
(seq ^\k)^\m=seq ^\(k + m);
theorem :: BHSP_3:32
(seq1 + seq2) ^\k = (seq1 ^\k) + (seq2 ^\k);
theorem :: BHSP_3:33
(-seq) ^\k = -(seq ^\k);
theorem :: BHSP_3:34
(seq1 - seq2) ^\k = (seq1 ^\k) - (seq2 ^\k);
theorem :: BHSP_3:35
(a * seq) ^\k = a * (seq ^\k);
theorem :: BHSP_3:36
seq is convergent implies (seq ^\k) is convergent & lim (seq ^\k)=lim
seq;
theorem :: BHSP_3:37
seq is convergent & (ex k st seq = seq1 ^\k) implies seq1 is convergent;
theorem :: BHSP_3:38
seq is Cauchy & (ex k st seq = seq1 ^\k) implies seq1 is Cauchy;
::$CT
theorem :: BHSP_3:40
seq1 is_compared_to seq2 implies (seq1 ^\k) is_compared_to (seq2 ^\k);
definition
let X;
attr X is complete means
:: BHSP_3:def 4
for seq holds seq is Cauchy implies seq is convergent;
end;
::$CT 2
theorem :: BHSP_3:43
X is complete & seq is Cauchy implies seq is bounded;