:: Cauchy Sequence of Complex Unitary Space
:: by Yasumasa Suzuki and Noboru Endou
::
:: Received March 18, 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, NAT_1, SEQ_1, COMSEQ_1, XCMPLX_0,
REAL_1, SUBSET_1, SUPINF_2, SERIES_1, ARYTM_3, FUNCT_1, CARD_1, ARYTM_1,
RELAT_1, COMPLEX1, SEQ_2, CARD_3, ORDINAL2, BHSP_3, XXREAL_0, NORMSP_1,
XXREAL_2, FUNCOP_1, VALUED_1, POWER, NEWTON, VALUED_0, INT_1, METRIC_1,
CSSPACE2;
notations SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XXREAL_0, COMPLEX1, XREAL_0,
REAL_1, FUNCOP_1, VALUED_1, SEQ_1, SEQ_2, INT_1, NAT_1, NEWTON, SERIES_1,
VALUED_0, PRE_TOPC, COMSEQ_1, STRUCT_0, RLVECT_1, VFUNCT_1, NORMSP_1,
BHSP_4, POWER, COMSEQ_2, COMSEQ_3, NORMSP_0, CLVECT_1, CSSPACE, CLVECT_2,
CSSPACE2;
constructors REAL_1, COMSEQ_2, COMSEQ_3, BHSP_4, CLVECT_2, SEQ_2, RELSET_1,
ABIAN, VFUNCT_1;
registrations ORDINAL1, FUNCT_2, XXREAL_0, XREAL_0, INT_1, MEMBERED, STRUCT_0,
VALUED_1, NUMBERS, VALUED_0, RELSET_1, FUNCOP_1, VFUNCT_1, XCMPLX_0,
COMSEQ_2, NEWTON, NAT_1;
requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM;
equalities RLVECT_1, VALUED_1;
expansions VALUED_1;
theorems NAT_1, FUNCT_2, SEQ_1, SEQ_2, SEQM_3, SERIES_1, ABSVALUE, RLVECT_1,
PREPOWER, POWER, INT_1, XCMPLX_1, CLVECT_1, CSSPACE, CLVECT_2, COMSEQ_1,
COMSEQ_2, COMPLEX1, COMSEQ_3, FUNCOP_1, XREAL_1, XXREAL_0, ORDINAL1,
BHSP_4, BHSP_1, NORMSP_1, VALUED_1;
schemes NAT_1, FUNCT_2;
begin :: Cauchy sequence of complex unitary space
reserve X for ComplexUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;
reserve Rseq for Real_Sequence;
reserve Cseq,Cseq1,Cseq2 for Complex_Sequence;
reserve z,z1,z2 for Complex;
reserve r for Real;
reserve k,n,m for Nat;
deffunc 09(ComplexUnitarySpace) = 0.$1;
theorem Th1:
Partial_Sums(seq1) + Partial_Sums(seq2) = Partial_Sums(seq1 + seq2)
proof
set PSseq1 = Partial_Sums(seq1);
set PSseq2 = Partial_Sums(seq2);
A1: now
let n;
thus (PSseq1 + PSseq2).(n + 1) = PSseq1.(n + 1) + PSseq2.(n + 1) by
NORMSP_1:def 2
.= PSseq1.n + seq1.(n + 1) + PSseq2.(n + 1) by BHSP_4:def 1
.= PSseq1.n + seq1.(n + 1) + (seq2.(n + 1) + PSseq2.n) by BHSP_4:def 1
.= PSseq1.n + seq1.(n + 1) + seq2.(n + 1) + PSseq2.n by RLVECT_1:def 3
.= PSseq1.n + (seq1.(n + 1) + seq2.(n + 1)) + PSseq2.n by RLVECT_1:def 3
.= PSseq1.n + (seq1 + seq2).(n + 1) + PSseq2.n by NORMSP_1:def 2
.= PSseq1.n + PSseq2.n + (seq1 + seq2).(n + 1) by RLVECT_1:def 3
.= (PSseq1 + PSseq2).n + (seq1 + seq2).(n + 1) by NORMSP_1:def 2;
end;
(PSseq1 + PSseq2).0 = PSseq1.0 + PSseq2.0 by NORMSP_1:def 2
.= seq1.0 + PSseq2.0 by BHSP_4:def 1
.= seq1.0 + seq2.0 by BHSP_4:def 1
.= (seq1 + seq2).0 by NORMSP_1:def 2;
hence thesis by A1,BHSP_4:def 1;
end;
theorem Th2:
Partial_Sums(seq1) - Partial_Sums(seq2) = Partial_Sums(seq1 - seq2)
proof
set PSseq1 = Partial_Sums(seq1);
set PSseq2 = Partial_Sums(seq2);
A1: now
let n;
thus (PSseq1 - PSseq2).(n + 1) = PSseq1.(n + 1) - PSseq2.(n + 1) by
NORMSP_1:def 3
.= (PSseq1.n + seq1.(n + 1)) - PSseq2.(n + 1) by BHSP_4:def 1
.= (PSseq1.n + seq1.(n + 1)) - (seq2.(n + 1) + PSseq2.n) by BHSP_4:def 1
.= ((PSseq1.n + seq1.(n + 1)) - seq2.(n + 1)) - PSseq2.n by RLVECT_1:27
.= (PSseq1.n + (seq1.(n + 1) - seq2.(n + 1))) - PSseq2.n by
RLVECT_1:def 3
.= (PSseq1.n - PSseq2.n) + (seq1.(n + 1) - seq2.(n + 1)) by
RLVECT_1:def 3
.= (PSseq1 - PSseq2).n + (seq1.(n + 1) - seq2.(n + 1)) by NORMSP_1:def 3
.= (PSseq1 - PSseq2).n + (seq1 - seq2).(n + 1) by NORMSP_1:def 3;
end;
(PSseq1 - PSseq2).0 = (PSseq1).0 - (PSseq2).0 by NORMSP_1:def 3
.= seq1.0 - (PSseq2).0 by BHSP_4:def 1
.= seq1.0 - seq2.0 by BHSP_4:def 1
.= (seq1 - seq2).0 by NORMSP_1:def 3;
hence thesis by A1,BHSP_4:def 1;
end;
theorem Th3:
Partial_Sums(z * seq) = z * Partial_Sums(seq)
proof
set PSseq = Partial_Sums(seq);
A1: now
let n;
thus (z * PSseq).(n + 1) = z * PSseq.(n + 1) by CLVECT_1:def 14
.= z * (PSseq.n + seq.(n + 1)) by BHSP_4:def 1
.= z * PSseq.n + z * seq.(n + 1) by CLVECT_1:def 2
.= (z * PSseq).n + z * seq.(n + 1) by CLVECT_1:def 14
.= (z * PSseq).n + (z * seq).(n + 1) by CLVECT_1:def 14;
end;
(z * PSseq).0 = z * PSseq.0 by CLVECT_1:def 14
.= z * seq.0 by BHSP_4:def 1
.= (z * seq).0 by CLVECT_1:def 14;
hence thesis by A1,BHSP_4:def 1;
end;
theorem
Partial_Sums(- seq) = - Partial_Sums(seq)
proof
Partial_Sums((-1r) * seq) = (-1r) * Partial_Sums(seq) by Th3;
then Partial_Sums(- seq) = (-1r) * Partial_Sums(seq) by CSSPACE:70;
hence thesis by CSSPACE:70;
end;
theorem
z1 * Partial_Sums(seq1) + z2 * Partial_Sums(seq2) = Partial_Sums(z1*
seq1 + z2*seq2)
proof
thus z1 * Partial_Sums(seq1) + z2 * Partial_Sums(seq2) = Partial_Sums(z1 *
seq1) + z2 * Partial_Sums(seq2) by Th3
.= Partial_Sums(z1 * seq1) + Partial_Sums(z2 * seq2) by Th3
.= Partial_Sums(z1 * seq1 + z2 * seq2) by Th1;
end;
definition
let X, seq;
attr seq is summable means
:Def1:
Partial_Sums(seq) is convergent;
func Sum(seq) -> Point of X equals
lim Partial_Sums(seq);
correctness;
end;
theorem
seq1 is summable & seq2 is summable implies seq1 + seq2 is summable &
Sum(seq1 + seq2) = Sum(seq1) + Sum(seq2)
proof
assume seq1 is summable & seq2 is summable;
then
A1: Partial_Sums(seq1) is convergent & Partial_Sums(seq2) is convergent;
then Partial_Sums(seq1) + Partial_Sums(seq2) is convergent by CLVECT_2:3;
then Partial_Sums(seq1 + seq2) is convergent by Th1;
hence seq1 + seq2 is summable;
thus Sum(seq1 + seq2) = lim (Partial_Sums(seq1) + Partial_Sums(seq2)) by Th1
.= Sum(seq1) + Sum(seq2) by A1,CLVECT_2:13;
end;
theorem
seq1 is summable & seq2 is summable implies seq1 - seq2 is summable &
Sum(seq1 - seq2) = Sum(seq1) - Sum(seq2)
proof
assume seq1 is summable & seq2 is summable;
then
A1: Partial_Sums(seq1) is convergent & Partial_Sums(seq2) is convergent;
then Partial_Sums(seq1) - Partial_Sums(seq2) is convergent by CLVECT_2:4;
then Partial_Sums(seq1 - seq2) is convergent by Th2;
hence seq1 - seq2 is summable;
thus Sum(seq1 - seq2) = lim (Partial_Sums(seq1) - Partial_Sums(seq2)) by Th2
.= Sum(seq1) - Sum(seq2) by A1,CLVECT_2:14;
end;
theorem
seq is summable implies z * seq is summable & Sum(z*seq) = z * Sum(seq )
proof
assume seq is summable;
then
A1: Partial_Sums(seq) is convergent;
then z * Partial_Sums(seq) is convergent by CLVECT_2:5;
then Partial_Sums(z * seq) is convergent by Th3;
hence z * seq is summable;
thus Sum(z * seq) = lim (z * Partial_Sums(seq)) by Th3
.= z * Sum(seq) by A1,CLVECT_2:15;
end;
theorem Th9:
seq is summable implies seq is convergent & lim seq = 0.X
proof
set PSseq = Partial_Sums(seq);
now
let n;
(PSseq).(n + 1) = (PSseq).n + seq.(n + 1) by BHSP_4:def 1
.= (PSseq).n + (seq ^\1).n by NAT_1:def 3;
hence (PSseq ^\1).n = (PSseq).n + (seq ^\1).n by NAT_1:def 3;
end;
then
A1: (PSseq ^\1) = PSseq + seq ^\1 by NORMSP_1:def 2;
now
let n be Element of NAT;
thus (seq ^\1 + (PSseq - PSseq)).n = (seq ^\1).n + (PSseq - PSseq).n by
NORMSP_1:def 2
.= (seq ^\1).n + (PSseq.n - PSseq.n) by NORMSP_1:def 3
.= (seq ^\1).n + 09(X) by RLVECT_1:15
.= (seq ^\1).n by RLVECT_1:4;
end;
then seq ^\1 + (PSseq - PSseq) = seq ^\1 by FUNCT_2:63;
then
A2: seq ^\1 = PSseq ^\1 - PSseq by A1,CSSPACE:76;
assume seq is summable;
then
A3: PSseq is convergent;
then
A4: PSseq ^\1 is convergent by CLVECT_2:90;
then
A5: seq ^\1 is convergent by A3,A2,CLVECT_2:4;
hence seq is convergent by CLVECT_2:91;
lim(PSseq ^\1) = lim(PSseq) by A3,CLVECT_2:90;
then lim(PSseq ^\1 - PSseq) = lim(PSseq) - lim(PSseq) by A3,A4,CLVECT_2:14
.= 09(X) by RLVECT_1:15;
hence thesis by A2,A5,CLVECT_2:84,91;
end;
theorem Th10:
for X being ComplexHilbertSpace, seq being sequence of X
holds seq is summable iff for r st r > 0 ex k
st for n, m st n >= k & m >= k holds ||.(Partial_Sums(seq)).n - (Partial_Sums(
seq)).m.|| < r
by CLVECT_2:65,CLVECT_2:58,CLVECT_2:def 11;
theorem
seq is summable implies Partial_Sums(seq) is bounded
by CLVECT_2:80;
theorem Th12:
(for n holds seq1.n = seq.0) implies Partial_Sums(seq^\1) = (
Partial_Sums(seq)^\1) - seq1
proof
assume
A1: for n holds seq1.n = seq.0;
A2: now
let n;
thus ((Partial_Sums(seq)^\1) - seq1).(n + 1) = (Partial_Sums(seq)^\1).(n +
1) - seq1.(n + 1) by NORMSP_1:def 3
.= (Partial_Sums(seq)^\1).(n + 1) - seq.0 by A1
.= Partial_Sums(seq).(n + 1 + 1) - seq.0 by NAT_1:def 3
.= seq.(n+1+1) + Partial_Sums(seq).(n + 1) - seq.0 by BHSP_4:def 1
.= seq.(n+1+1) + Partial_Sums(seq).(n + 1) - seq1.n by A1
.= seq.(n+1+1) + (Partial_Sums(seq).(n + 1) - seq1.n) by RLVECT_1:def 3
.= seq.(n+1+1) + ((Partial_Sums(seq)^\1).n - seq1.n) by NAT_1:def 3
.= seq.(n+1+1) + ((Partial_Sums(seq)^\1) - seq1).n by NORMSP_1:def 3
.= ((Partial_Sums(seq)^\1) - seq1).n + (seq^\1).(n + 1) by NAT_1:def 3;
end;
((Partial_Sums(seq)^\1) - seq1).0 = (Partial_Sums(seq)^\1).0 - seq1.0 by
NORMSP_1:def 3
.= (Partial_Sums(seq)^\1).0 - seq.0 by A1
.= Partial_Sums(seq).(0 + 1) - seq.0 by NAT_1:def 3
.= Partial_Sums(seq).0 + seq.(0 + 1) - seq.0 by BHSP_4:def 1
.= seq.(0 + 1) + seq.0 - seq.0 by BHSP_4:def 1
.= seq.(0 + 1) + (seq.0 - seq.0) by RLVECT_1:def 3
.= seq.(0 + 1) + 09(X) by RLVECT_1:15
.= seq.(0 + 1) by RLVECT_1:4
.= (seq^\1).0 by NAT_1:def 3;
hence thesis by A2,BHSP_4:def 1;
end;
theorem Th13:
seq is summable implies for k holds seq^\k is summable
proof
defpred P[Nat] means seq^\($1) is summable;
A1: for k st P[k] holds P[k+1]
proof
let k;
reconsider seq1 = NAT --> (seq^\k).0 as sequence of X;
assume seq^\k is summable;
then Partial_Sums(seq^\k) is convergent;
then
A2: Partial_Sums(seq^\k)^\1 is convergent by CLVECT_2:90;
for m holds seq1.m = (seq^\k).0
by ORDINAL1:def 12,FUNCOP_1:7;
then
seq1 is convergent & Partial_Sums(seq^\k^\1) = (Partial_Sums(seq^\k)^\
1) - seq1 by Th12,CLVECT_2:1;
then
seq^\(k+1)=(seq^\k)^\1 & Partial_Sums(seq^\k^\1) is convergent by A2,
CLVECT_2:4,NAT_1:48;
hence thesis by Def1;
end;
assume seq is summable;
then
A3: P[0] by NAT_1:47;
thus for k holds P[k] from NAT_1:sch 2(A3,A1);
end;
theorem
(ex k st seq^\k is summable) implies seq is summable
proof
given k such that
A1: seq^\k is summable;
seq^\k^\1 is summable by A1,Th13;
then
A2: Partial_Sums(seq^\k^\1) is convergent;
reconsider seq1 = NAT --> Partial_Sums(seq).k as sequence of X;
defpred P[Nat] means
(Partial_Sums(seq)^\(k+1)).$1 = Partial_Sums
(seq^\(k+1)).$1 + seq1.$1;
A3: Partial_Sums(seq^\(k+1)).0 = (seq^\(k+1)).0 by BHSP_4:def 1
.= seq.(0+(k+1)) by NAT_1:def 3
.= seq.(k+1);
A4: now
let m;
A5: m in NAT by ORDINAL1:def 12;
assume
A6: P[m];
Partial_Sums(seq^\(k+1)).(m+1) + seq1.(m+1) = seq1.(m+1) + (
Partial_Sums(seq^\(k+1)).m + (seq^\(k+1)).(m+1)) by BHSP_4:def 1
.= seq1.(m+1) + Partial_Sums(seq^\(k+1)).m + (seq^\(k+1)).(m+1) by
RLVECT_1:def 3
.= Partial_Sums(seq).k + Partial_Sums(seq^\(k+1)).m + (seq^\(k+1)).(m+
1) by FUNCOP_1:7
.= (Partial_Sums(seq)^\(k+1)).m + (seq^\(k+1)).(m+1) by A6,FUNCOP_1:7,A5
.= Partial_Sums(seq).(m+(k+1)) + (seq^\(k+1)).(m+1) by NAT_1:def 3
.= Partial_Sums(seq).(m+(k+1)) + seq.(m+1+(k+1)) by NAT_1:def 3
.= Partial_Sums(seq).(m+(k+1)+1) by BHSP_4:def 1
.= Partial_Sums(seq).(m+1+(k+1))
.= (Partial_Sums(seq)^\(k+1)).(m+1) by NAT_1:def 3;
hence P[m+1];
end;
(Partial_Sums(seq)^\(k+1)).0 = Partial_Sums(seq).(0+(k+1)) by NAT_1:def 3
.= Partial_Sums(seq).k + seq.(k+1) by BHSP_4:def 1
.= Partial_Sums(seq^\(k+1)).0 + seq1.0 by A3,FUNCOP_1:7;
then
A7: P[0];
for m holds P[m] from NAT_1:sch 2(A7,A4);
then
A8: Partial_Sums(seq)^\(k+1) = Partial_Sums(seq^\(k+1)) + seq1 by
NORMSP_1:def 2
.= Partial_Sums((seq^\k)^\1) + seq1 by NAT_1:48;
seq1 is convergent by CLVECT_2:1;
then Partial_Sums(seq)^\(k+1) is convergent by A2,A8,CLVECT_2:3;
then Partial_Sums(seq) is convergent by CLVECT_2:91;
hence thesis;
end;
definition
let X, seq, n;
func Sum(seq,n) -> Point of X equals
Partial_Sums(seq).n;
correctness;
end;
theorem
Sum(seq, 0) = seq.0 by BHSP_4:def 1;
theorem Th16:
Sum(seq,1) = Sum(seq,0) + seq.1
proof
Partial_Sums(seq).1 = Partial_Sums(seq).0 + seq.(0 + 1) by BHSP_4:def 1
.= Partial_Sums(seq).0 + seq.1;
hence thesis;
end;
theorem Th17:
Sum(seq,1) = seq.0 + seq.1
proof
thus Sum(seq,1) = Sum(seq,0) + seq.1 by Th16
.= seq.0 + seq.1 by BHSP_4:def 1;
end;
theorem
Sum(seq,n+1) = Sum(seq,n) + seq.(n+1) by BHSP_4:def 1;
theorem Th19:
seq.(n+1) = Sum(seq,n+1) - Sum(seq,n)
proof
thus Sum(seq,n+1)-Sum(seq,n) = (seq.(n+1)+Sum(seq,n)) - Sum(seq,n) by
BHSP_4:def 1
.= seq.(n+1) + (Sum(seq,n)-Sum(seq,n)) by RLVECT_1:def 3
.= seq.(n+1) + 0.X by RLVECT_1:15
.= seq.(n+1) by RLVECT_1:4;
end;
theorem
seq.1 = Sum(seq,1) - Sum(seq,0)
proof
seq.(0+1) = Sum(seq,0+1) - Sum(seq,0) by Th19;
hence thesis;
end;
definition
let X, seq, n, m;
func Sum(seq, n, m) -> Point of X equals
Sum(seq,n) - Sum(seq,m);
correctness;
end;
theorem
Sum(seq,1,0) = seq.1
proof
Sum(seq,1,0) = (seq.0 + seq.1) - Sum(seq,0) by Th17
.= (seq.1 + seq.0) - seq.0 by BHSP_4:def 1
.= seq.1 + (seq.0 - seq.0) by RLVECT_1:def 3
.= seq.1 + 09(X) by RLVECT_1:15;
hence thesis by RLVECT_1:4;
end;
theorem
Sum(seq,n+1,n) = seq.(n+1) by Th19;
theorem Th23:
for X being ComplexHilbertSpace, seq being sequence of X
holds seq is summable iff for r st r > 0 ex k
st for n, m st n >= k & m >= k holds ||.Sum(seq, n) - Sum(seq, m).|| < r
proof
let X be ComplexHilbertSpace, seq be sequence of X;
thus seq is summable implies for r st r > 0 ex k st for n, m st n >= k & m
>= k holds ||.Sum(seq, n) - Sum(seq, m).|| < r
proof
assume
A1: seq is summable;
now
let r;
assume r > 0;
then consider k such that
A2: for n, m st n >= k & m >= k holds ||.(Partial_Sums(seq)).n - (
Partial_Sums(seq)).m.|| < r by A1,Th10;
take k;
let n, m;
assume n >= k & m >= k;
hence ||.Sum(seq, n) - Sum(seq, m).|| < r by A2;
end;
hence thesis;
end;
thus ( for r st r > 0 ex k st for n, m st ( n >= k & m >= k ) holds ||.Sum(
seq, n) - Sum(seq, m).|| < r ) implies seq is summable
proof
assume
A3: for r st r > 0 ex k st for n, m st n >= k & m >= k holds ||.Sum(
seq, n) - Sum(seq, m).|| < r;
now
let r;
assume r > 0;
then consider k such that
A4: for n, m st n >= k & m >= k holds ||.Sum(seq, n) - Sum(seq, m)
.|| < r by A3;
take k;
let n, m;
assume n >= k & m >= k;
then ||.Sum(seq, n) - Sum(seq, m).|| < r by A4;
hence ||.(Partial_Sums(seq)).n - (Partial_Sums(seq)).m.|| < r;
end;
hence thesis by Th10;
end;
end;
theorem
for X being ComplexHilbertSpace, seq being sequence of X
holds seq is summable iff for r st r > 0 ex k st for
n, m st n>=k & m>=k holds ||.Sum(seq, n, m).|| < r
proof
let X be ComplexHilbertSpace, seq be sequence of X;
thus seq is summable implies for r st r > 0 ex k st for n, m st n >= k & m
>= k holds ||.Sum(seq, n, m).|| < r
proof
assume
A1: seq is summable;
now
let r;
assume r > 0;
then consider k such that
A2: for n, m st n >= k & m >= k holds ||.Sum(seq, n) - Sum(seq, m)
.|| < r by A1,Th23;
take k;
let n, m;
assume n >= k & m >= k;
hence ||.Sum(seq, n, m).|| < r by A2;
end;
hence thesis;
end;
thus ( for r st r > 0 ex k st for n, m st ( n >= k & m >= k ) holds ||.Sum(
seq, n, m).|| < r ) implies seq is summable
proof
assume
A3: for r st r > 0 ex k st for n, m st n >= k & m >= k holds ||.Sum(
seq, n, m).|| < r;
now
let r;
assume r > 0;
then consider k such that
A4: for n, m st n >= k & m >= k holds ||.Sum(seq, n, m).|| < r by A3;
take k;
let n, m;
assume n >= k & m >= k;
then ||.Sum(seq, n, m).|| < r by A4;
hence ||.Sum(seq, n) - Sum(seq, m).|| < r;
end;
hence thesis by Th23;
end;
end;
definition
let Cseq, n;
func Sum(Cseq,n) -> Complex equals
Partial_Sums(Cseq).n;
correctness;
end;
definition
let Cseq, n,m;
func Sum(Cseq,n,m) -> Complex equals
Sum(Cseq,n) - Sum(Cseq,m);
correctness;
end;
definition
let X, seq;
attr seq is absolutely_summable means
||.seq.|| is summable;
end;
theorem
seq1 is absolutely_summable & seq2 is absolutely_summable implies seq1
+ seq2 is absolutely_summable
proof
A1: for n holds ||.seq1 + seq2.||.n >= 0 & ||.seq1 + seq2.||.n <= (||.seq1
.|| + ||.seq2.||).n
proof
let n;
||.seq1 + seq2.||.n = ||.(seq1 + seq2).n.|| by CLVECT_2:def 3;
hence ||.seq1 + seq2.||.n >= 0 by CSSPACE:44;
||.seq1+seq2.||.n = ||.(seq1 + seq2).n.|| by CLVECT_2:def 3
.= ||.(seq1).n + (seq2).n.|| by NORMSP_1:def 2;
then ||.seq1+seq2.||.n <= ||.(seq1).n.|| + ||.(seq2).n.|| by CSSPACE:46;
then ||.seq1+seq2.||.n <= ||.seq1.||.n + ||.(seq2).n.|| by CLVECT_2:def 3;
then ||.seq1 + seq2.||.n <= ||.seq1.||.n + ||.seq2.||.n by CLVECT_2:def 3;
hence ||.seq1 + seq2.||.n <= (||.seq1.|| + ||.seq2.||).n by SEQ_1:7;
end;
assume seq1 is absolutely_summable & seq2 is absolutely_summable;
then ||.seq1.|| is summable & ||.seq2.|| is summable;
then ||.seq1.|| + ||.seq2.|| is summable by SERIES_1:7;
then ||.seq1 + seq2.|| is summable by A1,SERIES_1:20;
hence thesis;
end;
theorem
seq is absolutely_summable implies z * seq is absolutely_summable
proof
A1: for n holds ||.z * seq.||.n >= 0 & ||.z * seq.||.n <= (|.z.| (#) ||.seq
.||).n
proof
let n;
||.z * seq.||.n = ||.(z * seq).n.|| by CLVECT_2:def 3;
hence ||.z * seq.||.n >= 0 by CSSPACE:44;
||.z * seq.||.n = ||.(z * seq).n.|| by CLVECT_2:def 3
.= ||.z * seq.n.|| by CLVECT_1:def 14
.= |.z.| * ||.seq.n.|| by CSSPACE:43
.= |.z.| * ||.seq.||.n by CLVECT_2:def 3
.= (|.z.| (#) ||.seq.||).n by SEQ_1:9;
hence ||.z * seq.||.n <= (|.z.| (#) ||.seq.||).n;
end;
assume seq is absolutely_summable;
then ||.seq.|| is summable;
then |.z.| (#) ||.seq.|| is summable by SERIES_1:10;
then ||.z * seq.|| is summable by A1,SERIES_1:20;
hence thesis;
end;
theorem
( for n holds ||.seq.||.n <= Rseq.n ) & Rseq is summable implies seq
is absolutely_summable
proof
A1: for n holds ||.seq.||.n >= 0
proof
let n;
||.seq.||.n = ||.seq.n.|| by CLVECT_2:def 3;
hence thesis by CSSPACE:44;
end;
assume ( for n holds ||.seq.||.n <= Rseq.n)& Rseq is summable;
then ||.seq.|| is summable by A1,SERIES_1:20;
hence thesis;
end;
theorem
( for n holds seq.n <> 0.X & Rseq.n = ||.seq.(n+1).||/||.seq.n.|| ) &
Rseq is convergent & lim Rseq < 1 implies seq is absolutely_summable
proof
assume that
A1: for n holds seq.n <> 09(X) & Rseq.n = ||.seq.(n+1).||/||.seq.n.|| and
A2: Rseq is convergent & lim Rseq < 1;
for n holds ||.seq.||.n > 0 & Rseq.n = ||.seq.||.(n+1)/||.seq.||.n
proof
let n;
seq.n <> 09(X) by A1;
then
A3: ||.seq.n.|| <> 0 by CSSPACE:42;
||.seq.n.|| >= 0 by CSSPACE:44;
hence ||.seq.||.n > 0 by A3,CLVECT_2:def 3;
Rseq.n = ||.seq.(n+1).||/||.seq.n.|| by A1
.= ||.seq.||.(n+1)/||.seq.n.|| by CLVECT_2:def 3;
hence Rseq.n = ||.seq.||.(n+1)/||.seq.||.n by CLVECT_2:def 3;
end;
then ||.seq.|| is summable by A2,SERIES_1:26;
hence thesis;
end;
theorem Th29:
r > 0 & ( ex m st for n st n >= m holds ||.seq.n.|| >= r)
implies not seq is convergent or lim seq <> 0.X
proof
assume
A1: r > 0;
given m such that
A2: for n st n >= m holds ||.seq.n.|| >= r;
now
per cases;
suppose
not seq is convergent;
hence thesis;
end;
suppose
A3: seq is convergent;
now
assume lim seq = 09(X);
then consider k such that
A4: for n st n >= k holds ||.seq.n - 09(X).|| < r by A1,A3,CLVECT_2:19;
now
let n;
assume
A5: n >= m+k;
m+k >= k by NAT_1:11;
then n >= k by A5,XXREAL_0:2;
then ||.seq.n - 09(X).|| < r by A4;
then
A6: ||.seq.n.|| < r by RLVECT_1:13;
m+k >= m by NAT_1:11;
then n >= m by A5,XXREAL_0:2;
hence contradiction by A2,A6;
end;
hence contradiction;
end;
hence thesis;
end;
end;
hence thesis;
end;
theorem Th30:
( for n holds seq.n <> 0.X ) & ( ex m st for n st n >= m holds
||.seq.(n+1).||/||.seq.n.|| >= 1 ) implies not seq is summable
proof
assume
A1: for n holds seq.n <> 09(X);
given m such that
A2: for n st n >= m holds ||.seq.(n+1).||/||.seq.n.|| >= 1;
A3: now
defpred P[Nat] means ||.seq.(m+$1).|| >= ||.seq.m.||;
let n;
A4: for k st P[k] holds P[k+1]
proof
let k;
assume
A5: ||.seq.(m+k).|| >= ||.seq.m.||;
seq.(m+k) <> 09(X) by A1;
then
A6: ||.seq.(m+k).|| <> 0 by CSSPACE:42;
||.seq.(m+k+1).||/||.seq.(m+k).|| >= 1 & ||.seq.(m+k).|| >= 0 by A2,
CSSPACE:44,NAT_1:11;
then ||.seq.(m+k+1).|| >= ||.seq.(m+k).|| by A6,XREAL_1:191;
hence thesis by A5,XXREAL_0:2;
end;
A7: P[0];
A8: for k holds P[k] from NAT_1:sch 2(A7,A4);
assume n >= m;
then consider k being Nat such that
A9: n = m + k by NAT_1:10;
thus ||.seq.n.|| >= ||.seq.m.|| by A8,A9;
end;
seq.m <> 09(X) by A1;
then ||.seq.m.|| <> 0 by CSSPACE:42;
then ||.seq.m.|| > 0 by CSSPACE:44;
then not seq is convergent or lim seq <> 09(X) by A3,Th29;
hence thesis by Th9;
end;
theorem
(for n holds seq.n <> 0.X) & (for n holds Rseq.n = ||.seq.(n+1).||/||.
seq.n.||) & Rseq is convergent & lim Rseq > 1 implies not seq is summable
proof
assume that
A1: for n holds seq.n <> 09(X) and
A2: for n holds Rseq.n = ||.seq.(n+1).||/||.seq.n.|| and
A3: Rseq is convergent and
A4: lim Rseq > 1;
lim Rseq - 1 > 0 by A4,XREAL_1:50;
then consider m such that
A5: for n st n >= m holds |.Rseq.n - lim Rseq.| < lim Rseq - 1 by A3,
SEQ_2:def 7;
now
let n;
A6: m + 1 >= m by NAT_1:11;
A7: Rseq.n = ||.seq.(n+1).||/||.seq.n.|| by A2;
assume n >= m + 1;
then n >= m by A6,XXREAL_0:2;
then
|.||.seq.(n+1).||/||.seq.n.|| - lim Rseq.| < lim Rseq - 1 by A5,A7;
then - (lim Rseq - 1) < ||.seq.(n+1).||/||.seq.n.|| - lim Rseq by SEQ_2:1;
then 1 - lim Rseq + lim Rseq < ||.seq.(n+1).||/||.seq.n.|| - lim Rseq+lim
Rseq by XREAL_1:6;
hence ||.seq.(n+1).||/||.seq.n.|| >= 1;
end;
hence thesis by A1,Th30;
end;
theorem
( for n holds Rseq.n = n-root (||.seq.n.||) ) & Rseq is convergent &
lim Rseq < 1 implies seq is absolutely_summable
proof
assume that
A1: for n holds Rseq.n = n-root (||.seq.n.||) and
A2: Rseq is convergent & lim Rseq < 1;
for n holds ||.seq.||.n >= 0 & Rseq.n = n-root (||.seq.||.n)
proof
let n;
||.seq.||.n = ||.seq.n.|| by CLVECT_2:def 3;
hence ||.seq.||.n >= 0 by CSSPACE:44;
Rseq.n = n-root (||.seq.n.||) by A1;
hence Rseq.n = n-root (||.seq.||.n) by CLVECT_2:def 3;
end;
then ||.seq.|| is summable by A2,SERIES_1:28;
hence thesis;
end;
theorem
(for n holds Rseq.n = n-root (||.seq.||.n)) & (ex m st for n st n >= m
holds Rseq.n >= 1) implies not seq is summable
proof
assume
A1: for n holds Rseq.n = n-root (||.seq.||.n);
given m such that
A2: for n st n >= m holds Rseq.n >= 1;
now
let n;
assume
A3: n >= m+1;
m + 1 >= 1 by NAT_1:11;
then
A4: n >= 1 by A3,XXREAL_0:2;
m+1 >= m by NAT_1:11;
then
A5: n>=m by A3,XXREAL_0:2;
Rseq.n = n-root (||.seq.||.n) by A1
.= n-root ||.seq.n.|| by CLVECT_2:def 3;
then ||.seq.n.|| >= 0 & n-root ||.seq.n.|| |^ n >= 1 by A2,A5,CSSPACE:44
,PREPOWER:11;
hence ||.seq.n.|| >= 1 by A4,POWER:4;
end;
then not seq is convergent or lim seq <> 09(X) by Th29;
hence thesis by Th9;
end;
theorem
(for n holds Rseq.n = n-root (||.seq.||.n)) & Rseq is convergent & lim
Rseq > 1 implies not seq is summable
proof
assume that
A1: for n holds Rseq.n = n-root (||.seq.||.n) and
A2: Rseq is convergent and
A3: lim Rseq > 1;
lim Rseq - 1 > 0 by A3,XREAL_1:50;
then consider m such that
A4: for n st n >= m holds |.Rseq.n - lim Rseq.| < lim Rseq - 1 by A2,
SEQ_2:def 7;
now
let n;
assume
A5: n >= m + 1;
A6: Rseq.n = n-root (||.seq.||.n) by A1
.= n-root ||.seq.n.|| by CLVECT_2:def 3;
m + 1 >= m by NAT_1:11;
then n >= m by A5,XXREAL_0:2;
then |.n-root ||.seq.n.|| - lim Rseq.| < lim Rseq - 1 by A4,A6;
then - (lim Rseq - 1) < n-root ||.seq.n.|| - lim Rseq by SEQ_2:1;
then
1 - lim Rseq + lim Rseq < n-root ||.seq.n.|| - lim Rseq + lim Rseq by
XREAL_1:6;
then
A7: ||.seq.n.|| >= 0 & n-root ||.seq.n.|| |^ n >= 1 by CSSPACE:44,PREPOWER:11;
m + 1 >= 1 by NAT_1:11;
then n >= 1 by A5,XXREAL_0:2;
hence ||.seq.n.|| >= 1 by A7,POWER:4;
end;
then not seq is convergent or lim seq <> 09(X) by Th29;
hence thesis by Th9;
end;
theorem Th35:
Partial_Sums(||.seq.||) is non-decreasing
proof
now
let n;
||.seq.(n+1).|| >= 0 by CSSPACE:44;
then ||.seq.||.(n+1) >= 0 by CLVECT_2:def 3;
then
0 + Partial_Sums(||.seq.||).n <= ||.seq.||.(n+1) + Partial_Sums(||.seq
.||).n by XREAL_1:6;
hence Partial_Sums(||.seq.||).n <= Partial_Sums(||.seq.||).(n+1) by
SERIES_1:def 1;
end;
hence thesis;
end;
theorem
for n holds Partial_Sums(||.seq.||).n >= 0
proof
let n;
||.(seq.0).|| >= 0 by CSSPACE:44;
then ||.seq.||.0 >= 0 by CLVECT_2:def 3;
then Partial_Sums(||.seq.||).0 >= 0 by SERIES_1:def 1;
hence thesis by Th35,SEQM_3:11;
end;
theorem Th37:
for n holds ||.Partial_Sums(seq).n.|| <= Partial_Sums(||.seq.||) .n
proof
defpred P[Nat] means
||.Partial_Sums(seq).$1.|| <= Partial_Sums(||.seq.||).$1;
A1: now
let n;
Partial_Sums(seq).(n+1) = Partial_Sums(seq).n + seq.(n+1) by BHSP_4:def 1;
then
A2: ||.Partial_Sums(seq).(n+1).|| <= ||.Partial_Sums(seq).n.|| + ||.seq.(n
+1).|| by CSSPACE:46;
assume P[n];
then
||.Partial_Sums(seq).n.|| + ||.seq.(n+1).|| <= Partial_Sums(||.seq.||)
.n + ||.seq.(n+1).|| by XREAL_1:6;
then
||.Partial_Sums(seq).(n+1).|| <= Partial_Sums(||.seq.||).n + ||.seq.(n
+1).|| by A2,XXREAL_0:2;
then
||.Partial_Sums(seq).(n+1).|| <= Partial_Sums(||.seq.||).n + ||.seq.||
.(n+1) by CLVECT_2:def 3;
hence P[n+1] by SERIES_1:def 1;
end;
Partial_Sums(||.seq.||).0 = ||.seq.||.0 by SERIES_1:def 1
.= ||.(seq.0).|| by CLVECT_2:def 3;
then
A3: P[0] by BHSP_4:def 1;
thus for n holds P[n] from NAT_1:sch 2(A3,A1);
end;
theorem
for n holds ||.Sum(seq, n).|| <= Sum(||.seq.||, n)
proof
let n;
||.Partial_Sums(seq).n.|| <= Partial_Sums(||.seq.||).n by Th37;
hence thesis by SERIES_1:def 5;
end;
theorem Th39:
for n, m holds ||.Partial_Sums(seq).m - Partial_Sums(seq).n.||
<= |.Partial_Sums(||.seq.||).m - Partial_Sums(||.seq.||).n.|
proof
let n, m;
A1: now
reconsider d = n, t = m as Integer;
set PSseq9 = Partial_Sums(||.seq.||);
set PSseq = Partial_Sums(seq);
defpred P[Nat] means
||.PSseq.m - PSseq.(m+$1).|| <= |.PSseq9
.m - PSseq9.(m+$1).|;
A2: PSseq9 is non-decreasing by Th35;
A3: now
let k;
A4: ||.seq.(m+k+1).|| >= 0 by CSSPACE:44;
A5: |.PSseq9.m - PSseq9.(m+(k+1)).| = |.-(PSseq9.(m+(k+1)) - PSseq9. m).|
.= |.PSseq9.(m+k+1) - PSseq9.m.| by COMPLEX1:52
.= |.PSseq9.(m+k) + (||.seq.||).(m+k+1) - PSseq9.m.| by SERIES_1:def 1
.= |.||.seq.(m+k+1).|| + PSseq9.(m+k) - PSseq9.m.| by CLVECT_2:def 3
.= |.(PSseq9.(m+k) - PSseq9.m) + ||.seq.(m+k+1).||.|;
||.PSseq.m - PSseq.(m+(k+1)).|| = ||.PSseq.m - (PSseq.(m+k) + seq.(
m+k+1)).|| by BHSP_4:def 1
.= ||.(PSseq.m - PSseq.(m+k)) - seq.(m+k+1).|| by RLVECT_1:27
.= ||.(PSseq.m - PSseq.(m+k)) + -seq.(m+k+1).||;
then ||.PSseq.m - PSseq.(m+(k+1)).|| <= ||.PSseq.m - PSseq.(m+k).|| +
||.-seq.(m+k+1).|| by CSSPACE:46;
then
A6: ||.PSseq.m - PSseq.(m+(k+1)).|| <= ||.PSseq.m - PSseq.(m+k).|| +
||.seq.(m+k+1).|| by CSSPACE:47;
PSseq9.(m+k) >= PSseq9.m by A2,SEQM_3:5;
then
A7: PSseq9.(m+k) - PSseq9.m >= 0 by XREAL_1:48;
assume P[k];
then ||.PSseq.m - PSseq.(m+k).|| + ||.seq.(m+k+1).|| <= |.PSseq9.m-
PSseq9.(m+k).| +||.seq.(m+k+1).|| by XREAL_1:6;
then ||.PSseq.m - PSseq.(m+(k+1)).|| <= |.-(PSseq9.(m+k) - PSseq9.m).|
+ ||.seq.(m+k+1).|| by A6,XXREAL_0:2;
then ||.PSseq.m - PSseq.(m+(k+1)).|| <= |.PSseq9.(m+k) - PSseq9.m.| +
||.seq.(m+k+1).|| by COMPLEX1:52;
then ||.PSseq.m - PSseq.(m+(k+1)).|| <= (PSseq9.(m+k) - PSseq9.m) + ||.
seq.(m+k+1).|| by A7,ABSVALUE:def 1;
hence P[k+1] by A7,A4,A5,ABSVALUE:def 1;
end;
assume n >= m;
then reconsider k = d - t as Element of NAT by INT_1:5;
A8: m + k = n;
||.PSseq.m - PSseq.(m+0).|| = ||.09(X).|| by RLVECT_1:15
.= 0 by CSSPACE:42;
then
A9: P[0] by COMPLEX1:46;
for k holds P[k] from NAT_1:sch 2(A9,A3);
hence thesis by A8;
end;
now
reconsider d = n, t = m as Integer;
set PSseq9 = Partial_Sums(||.seq.||);
set PSseq = Partial_Sums(seq);
defpred P[Nat] means
||.PSseq.(n+$1) - PSseq.n.|| <= |.PSseq9
.(n+$1) - PSseq9.n.|;
A10: PSseq9 is non-decreasing by Th35;
A11: now
let k;
A12: |.PSseq9.(n+(k+1)) - PSseq9.n.| = |.PSseq9.(n+k) + (||.seq.||).(
n+k+1) - PSseq9.n.| by SERIES_1:def 1
.= |.||.seq.(n+k+1).|| + PSseq9.(n+k) - PSseq9.n.| by CLVECT_2:def 3
.= |.||.seq.(n+k+1).|| + (PSseq9.(n+k) - PSseq9.n).|;
||.PSseq.(n+(k+1)) - PSseq.n.|| = ||.seq.(n+k+1) + PSseq.(n+k) -
PSseq.n.|| by BHSP_4:def 1
.= ||.seq.(n+k+1) + (PSseq.(n+k) - PSseq.n).|| by RLVECT_1:def 3;
then
A13: ||.PSseq.(n+(k+1)) - PSseq.n.|| <= ||.seq.(n+k+1).|| + ||.PSseq.(n+k
) - PSseq.n.|| by CSSPACE:46;
PSseq9.(n+k) >= PSseq9.n by A10,SEQM_3:5;
then
A14: PSseq9.(n+k) - PSseq9.n >= 0 by XREAL_1:48;
assume P[k];
then ||.seq.(n+k+1).|| + ||.PSseq.(n+k) - PSseq.n.|| <= ||.seq.(n+k+1)
.|| + |.PSseq9.(n+k)-PSseq9.n.| by XREAL_1:7;
then
||.PSseq.(n+(k+1)) - PSseq.n.|| <= ||.seq.(n+k+1).|| + |.PSseq9.(
n+k)-PSseq9.n.| by A13,XXREAL_0:2;
then
A15: ||.PSseq.(n+(k+1)) - PSseq.n.|| <= ||.seq.(n+k+1).|| + (PSseq9.(n+k
)-PSseq9.n) by A14,ABSVALUE:def 1;
||.seq.(n+k+1).|| >= 0 by CSSPACE:44;
hence P[k+1] by A14,A15,A12,ABSVALUE:def 1;
end;
assume n <= m;
then reconsider k = t - d as Element of NAT by INT_1:5;
A16: n + k = m;
||.PSseq.(n+0) - PSseq.n.|| = ||.09(X).|| by RLVECT_1:15
.= 0 by CSSPACE:42;
then
A17: P[0] by COMPLEX1:46;
for k holds P[k] from NAT_1:sch 2(A17,A11);
hence thesis by A16;
end;
hence thesis by A1;
end;
theorem Th40:
for n,m holds ||.Sum(seq,m)-Sum(seq,n).|| <= |. Sum(||.seq.||,
m)-Sum(||.seq.||,n) .|
proof
let n,m;
||.Sum(seq, m) - Partial_Sums(seq).n.|| <= |.Partial_Sums(||.seq.||).m
- Partial_Sums(||.seq.||).n.| by Th39;
then
||.Sum(seq, m) - Sum(seq, n).|| <= |.Sum(||.seq.||, m) - Partial_Sums(
||.seq.||).n.| by SERIES_1:def 5;
hence thesis by SERIES_1:def 5;
end;
theorem
for n,m holds ||.Sum(seq,m,n).|| <= |.Sum(||.seq.||,m,n).|
proof
let n,m;
||.Sum(seq,m) - Sum(seq,n).|| <= |.Sum(||.seq.||, m) - Sum(||.seq.||,
n).| by Th40;
hence thesis by SERIES_1:def 6;
end;
theorem
for X being ComplexHilbertSpace, seq being sequence of X
holds seq is absolutely_summable implies seq is
summable
proof
let X be ComplexHilbertSpace, seq be sequence of X such that
A1: seq is absolutely_summable;
A2: ||.seq.|| is summable by A1;
now
let r;
assume r > 0;
then r/2 > 0;
then consider k such that
A3: for m st m >= k holds |.Partial_Sums(||.seq.||).m - Partial_Sums
(||.seq.||).k.| < r/2 by A2,SERIES_1:21;
take k;
now
let m, n;
A4: ||.Partial_Sums(seq).n - Partial_Sums(seq).m.|| <= |.Partial_Sums
(||.seq.||).n - Partial_Sums(||.seq.||).m.| by Th39;
assume m >= k & n >= k;
then
|.Partial_Sums(||.seq.||).m - Partial_Sums(||.seq.||).k.| < r /2 &
|. Partial_Sums(||.seq.||).n - Partial_Sums(||.seq.||).k.| < r/2 by A3;
then
A5: |.Partial_Sums(||.seq.||).n - Partial_Sums(||.seq.||).k.| + |.
Partial_Sums(||.seq.||).m - Partial_Sums(||.seq.||).k.| < r/2 + r/2 by
XREAL_1:8;
|.Partial_Sums(||.seq.||).n - Partial_Sums(||.seq.||).m.| = |.(
Partial_Sums(||.seq.||).n - Partial_Sums(||.seq.||).k) - (Partial_Sums(||.seq
.||).m - Partial_Sums(||.seq.||).k).|;
then |.Partial_Sums(||.seq.||).n - Partial_Sums(||.seq.||).m.| <= |.
Partial_Sums(||.seq.||).n - Partial_Sums(||.seq.||).k.| + |.Partial_Sums(||.
seq.||).m - Partial_Sums(||.seq.||).k.| by COMPLEX1:57;
then |.Partial_Sums(||.seq.||).n - Partial_Sums(||.seq.||).m.| < r by A5
,XXREAL_0:2;
hence ||.Partial_Sums(seq).n - Partial_Sums(seq).m.|| < r by A4,
XXREAL_0:2;
end;
hence for n, m st n >= k & m >= k holds ||.Partial_Sums(seq).n -
Partial_Sums(seq).m.|| < r;
end;
hence thesis by Th10;
end;
definition
let X, seq, Cseq;
func Cseq * seq -> sequence of X means
:Def8:
for n holds it.n = Cseq.n * seq.n;
existence
proof
deffunc F(Nat) = Cseq.$1 * seq.$1;
consider M being sequence of X such that
A1: for n being Element of NAT holds M.n = F(n)
from FUNCT_2:sch 4;
take M;
let n be Nat;
n in NAT by ORDINAL1:def 12;
hence thesis by A1;
end;
uniqueness
proof
let seq1, seq2;
assume that
A2: for n holds seq1.n = Cseq.n * seq.n and
A3: for n holds seq2.n = Cseq.n * seq.n;
now
let n be Element of NAT;
seq1.n = Cseq.n * seq.n by A2;
hence seq1.n = seq2.n by A3;
end;
hence thesis by FUNCT_2:63;
end;
end;
theorem
Cseq * (seq1 + seq2) = Cseq * seq1 + Cseq * seq2
proof
now
let n be Element of NAT;
thus (Cseq * (seq1 + seq2)).n = Cseq.n * (seq1 + seq2).n by Def8
.= Cseq.n * (seq1.n + seq2.n) by NORMSP_1:def 2
.= Cseq.n * seq1.n + Cseq.n * seq2.n by CLVECT_1:def 2
.= (Cseq * seq1).n + Cseq.n * seq2.n by Def8
.= (Cseq * seq1).n + (Cseq * seq2).n by Def8
.= (Cseq * seq1 + Cseq * seq2).n by NORMSP_1:def 2;
end;
hence thesis by FUNCT_2:63;
end;
theorem
(Cseq1 + Cseq2) * seq = Cseq1 * seq + Cseq2 * seq
proof
now
let n be Element of NAT;
thus ((Cseq1 + Cseq2) * seq).n = (Cseq1 + Cseq2).n * seq.n by Def8
.= (Cseq1.n + Cseq2.n) * seq.n by VALUED_1:1
.= Cseq1.n * seq.n + Cseq2.n * seq.n by CLVECT_1:def 3
.= (Cseq1 * seq).n + Cseq2.n * seq.n by Def8
.= (Cseq1 * seq).n + (Cseq2 * seq).n by Def8
.= (Cseq1 * seq + Cseq2 * seq).n by NORMSP_1:def 2;
end;
hence thesis by FUNCT_2:63;
end;
theorem
(Cseq1 (#) Cseq2) * seq = Cseq1 * (Cseq2 * seq)
proof
now
let n be Element of NAT;
thus ((Cseq1 (#) Cseq2) * seq).n = (Cseq1 (#) Cseq2).n * seq.n by Def8
.= (Cseq1.n * Cseq2.n) * seq.n by VALUED_1:5
.= Cseq1.n * (Cseq2.n * seq.n) by CLVECT_1:def 4
.= Cseq1.n * (Cseq2 * seq).n by Def8
.= (Cseq1 * (Cseq2 * seq)).n by Def8;
end;
hence thesis by FUNCT_2:63;
end;
theorem Th46:
(z (#) Cseq) * seq = z * (Cseq * seq)
proof
now
let n be Element of NAT;
thus ((z (#) Cseq) * seq).n = (z (#) Cseq).n * seq.n by Def8
.= (z* Cseq.n) * seq.n by VALUED_1:6
.= z * (Cseq.n * seq.n) by CLVECT_1:def 4
.= z * (Cseq * seq).n by Def8
.= (z * (Cseq * seq)).n by CLVECT_1:def 14;
end;
hence thesis by FUNCT_2:63;
end;
theorem
Cseq * (- seq) = (- Cseq) * seq
proof
now
let n be Element of NAT;
thus (Cseq * (- seq)).n = Cseq.n * (-seq).n by Def8
.= Cseq.n * (-(seq.n)) by BHSP_1:44
.= (-(Cseq.n)) * seq.n by CLVECT_1:6
.= (- Cseq).n * seq.n by VALUED_1:8
.= ((- Cseq) * seq).n by Def8;
end;
hence thesis by FUNCT_2:63;
end;
theorem Th48:
Cseq is convergent & seq is convergent implies Cseq * seq is convergent
proof
assume that
A1: Cseq is convergent and
A2: seq is convergent;
consider p being Complex such that
A3: for r being Real st r > 0 ex m st for n st n >= m holds |.(Cseq.n -
p).| < r by A1,COMSEQ_2:def 5;
consider g such that
A4: for r st r > 0 ex m st for n st n >= m holds ||.seq.n - g.|| < r by A2,
CLVECT_2:9;
now
take h = p * g;
let r;
Cseq is bounded by A1;
then consider b being Real such that
A5: b > 0 and
A6: for n holds |.(Cseq.n).| < b by COMSEQ_2:8;
A7: b + ||.g.|| > 0 + 0 by A5,CSSPACE:44,XREAL_1:8;
assume
A8: r > 0;
then consider m1 be Nat such that
A9: for n st n >= m1 holds |.(Cseq.n - p).| < r/(b + ||.g.||) by A3,A7;
consider m2 be Nat such that
A10: for n st n >= m2 holds ||.seq.n - g.|| < r/(b + ||.g.||) by A4,A7,A8;
take m = m1 + m2;
let n such that
A11: n >= m;
m1 + m2 >= m1 by NAT_1:12;
then n >= m1 by A11,XXREAL_0:2;
then ||.g.|| >= 0 & |.(Cseq.n - p).| <= r/(b + ||.g.||) by A9,CSSPACE:44;
then
A12: ||.g.|| * |.(Cseq.n - p).| <= ||.g.|| * (r/(b + ||.g.||)) by XREAL_1:64;
A13: |.(Cseq.n).| >= 0 & ||.seq.n - g.|| >= 0 by COMPLEX1:46,CSSPACE:44;
m >= m2 by NAT_1:12;
then n >= m2 by A11,XXREAL_0:2;
then
A14: ||.seq.n - g.|| < r/(b + ||.g.||) by A10;
|.(Cseq.n).| < b by A6;
then |.(Cseq.n).| * ||.seq.n - g.|| < b * (r/(b + ||.g.||)) by A14,A13,
XREAL_1:96;
then
|.(Cseq.n).| * ||.seq.n - g.|| + ||.g.|| * |.(Cseq.n - p).| < b*(r/(b
+ ||.g.||)) + ||.g.||*(r/(b + ||.g.||)) by A12,XREAL_1:8;
then
|.(Cseq.n).| * ||.seq.n - g.|| + ||.g.|| * |.(Cseq.n - p).| < (b*r)/(
b + ||.g.||) + ||.g.||*(r/(b + ||.g.||)) by XCMPLX_1:74;
then
|.(Cseq.n).| * ||.seq.n - g.|| + ||.g.|| * |.(Cseq.n - p).| < (b*r)/(
b + ||.g.||) + (||.g.||*r)/(b + ||.g.||) by XCMPLX_1:74;
then |.(Cseq.n).| * ||.seq.n - g.|| + ||.g.|| * |.(Cseq.n - p).| < (b * r
+ ||.g.|| * r)/(b + ||.g.||) by XCMPLX_1:62;
then |.(Cseq.n).| * ||.seq.n - g.|| + ||.g.|| * |.(Cseq.n - p).| < ((b +
||.g.||) * r)/(b + ||.g.||);
then
A15: |.(Cseq.n).| * ||.seq.n - g.|| + ||.g.|| * |.(Cseq.n - p).| < r by A7,
XCMPLX_1:89;
||.(Cseq * seq).n - p * g.|| = ||.Cseq.n * seq.n - p * g.|| by Def8
.= ||.(Cseq.n * seq.n - p * g) + 09(X).|| by RLVECT_1:4
.= ||.(Cseq.n * seq.n - p * g) + (Cseq.n * g - Cseq.n * g).|| by
RLVECT_1:15
.= ||.Cseq.n * seq.n - (p * g - (Cseq.n * g - Cseq.n * g)).|| by
RLVECT_1:29
.= ||.Cseq.n * seq.n - (Cseq.n * g + (p * g - Cseq.n * g)).|| by
RLVECT_1:29
.= ||.(Cseq.n * seq.n - Cseq.n * g) - (p * g - Cseq.n * g).|| by
RLVECT_1:27
.= ||.(Cseq.n * seq.n - Cseq.n * g) + (Cseq.n * g - p * g).|| by
RLVECT_1:33;
then ||.(Cseq * seq).n - p * g.|| <= ||.Cseq.n * seq.n - Cseq.n * g.|| +
||.Cseq.n * g - p * g.|| by CSSPACE:46;
then
||.(Cseq * seq).n - p * g.|| <= ||.Cseq.n * (seq.n - g).|| + ||.Cseq.
n * g - p * g.|| by CLVECT_1:9;
then
||.(Cseq * seq).n - p * g.|| <= ||.Cseq.n * (seq.n - g).|| + ||.(Cseq
.n - p)*g.|| by CLVECT_1:10;
then
||.(Cseq * seq).n - p * g.|| <= |.(Cseq.n).| * ||.seq.n - g.|| + ||.(
Cseq.n - p) * g.|| by CSSPACE:43;
then
||.(Cseq * seq).n - p * g.|| <= |.(Cseq.n).| * ||.seq.n - g.|| + ||.g
.|| * |.(Cseq.n - p).| by CSSPACE:43;
hence ||.(Cseq * seq).n - h.|| < r by A15,XXREAL_0:2;
end;
hence thesis by CLVECT_2:9;
end;
theorem
Cseq is bounded & seq is bounded implies Cseq * seq is bounded
proof
assume that
A1: Cseq is bounded and
A2: seq is bounded;
consider M1 being Real such that
A3: M1 > 0 and
A4: for n holds |.(Cseq.n).| < M1 by A1,COMSEQ_2:8;
consider M2 be Real such that
A5: M2 > 0 and
A6: for n holds ||.seq.n.|| <= M2 by A2,CLVECT_2:def 10;
now
set M = M1 * M2;
take M;
now
let n;
|.(Cseq.n).| >= 0 by COMPLEX1:46;
then
A7: |.(Cseq.n).| * ||.seq.n.|| <= |.(Cseq.n).| * M2 by A6,XREAL_1:64;
|.(Cseq.n).| <= M1 by A4;
then
A8: |.(Cseq.n).| * M2 <= M1 * M2 by A5,XREAL_1:64;
||.(Cseq * seq).n.|| = ||.Cseq.n * seq.n.|| by Def8
.= |.(Cseq.n).| * ||.seq.n.|| by CSSPACE:43;
hence ||.(Cseq * seq).n.|| <= M by A7,A8,XXREAL_0:2;
end;
hence M > 0 & for n holds ||.(Cseq * seq).n.|| <= M by A3,A5;
end;
hence thesis by CLVECT_2:def 10;
end;
theorem
Cseq is convergent & seq is convergent implies Cseq * seq is
convergent & lim (Cseq * seq) = lim Cseq * lim seq
proof
assume that
A1: Cseq is convergent and
A2: seq is convergent;
Cseq is bounded by A1;
then consider b being Real such that
A3: b > 0 and
A4: for n holds |.(Cseq.n).| < b by COMSEQ_2:8;
A5: b + ||.lim seq.|| > 0 + 0 by A3,CSSPACE:44,XREAL_1:8;
A6: ||.lim seq.|| >= 0 by CSSPACE:44;
A7: now
let r;
assume r > 0;
then
A8: r/(b + ||.lim seq.||) > 0 by A5;
then consider m1 be Nat such that
A9: for n st n >= m1 holds |.(Cseq.n - lim Cseq).| < r/(b + ||.lim
seq.||) by A1,COMSEQ_2:def 6;
consider m2 be Nat such that
A10: for n st n >= m2 holds dist(seq.n, lim seq) < r/(b + ||.lim seq
.||) by A2,A8,CLVECT_2:def 2;
take m = m1 + m2;
let n such that
A11: n >= m;
m1 + m2 >= m1 by NAT_1:12;
then n >= m1 by A11,XXREAL_0:2;
then |.(Cseq.n - lim Cseq).| <= r/(b + ||.lim seq.||) by A9;
then
A12: ||.lim seq.|| * |.(Cseq.n - lim Cseq).| <= ||.lim seq.|| * (r/(b +
||.lim seq.||)) by A6,XREAL_1:64;
A13: ||.seq.n - lim seq.|| >= 0 by CSSPACE:44;
m >= m2 by NAT_1:12;
then n >= m2 by A11,XXREAL_0:2;
then dist(seq.n, lim seq) < r/(b + ||.lim seq.||) by A10;
then
A14: ||.seq.n - lim seq.|| < r/(b + ||.lim seq.||) by CSSPACE:def 16;
|.(Cseq.n).| < b & |.(Cseq.n).| >= 0 by A4,COMPLEX1:46;
then |.(Cseq.n).| * ||.seq.n - lim seq.|| < b * (r/(b + ||.lim seq.||) )
by A14,A13,XREAL_1:96;
then |.(Cseq.n).|*||.seq.n-lim seq.|| + ||.lim seq.||*|.(Cseq.n-lim Cseq)
.| < b * (r/(b + ||.lim seq.||)) + ||.lim seq.|| * (r/(b + ||.lim seq.||)) by
A12,XREAL_1:8;
then |.(Cseq.n).| * ||.seq.n - lim seq.|| + ||.lim seq.|| * |.(Cseq.n -
lim Cseq).| < (b * r)/(b + ||.lim seq.||) + ||.lim seq.|| * (r/(b + ||.lim seq
.||)) by XCMPLX_1:74;
then |.(Cseq.n).| * ||.seq.n - lim seq.|| + ||.lim seq.|| * |.(Cseq.n -
lim Cseq).| < (b * r)/(b + ||.lim seq.||) + (||.lim seq.|| * r)/(b + ||.lim seq
.||) by XCMPLX_1:74;
then |.(Cseq.n).| * ||.seq.n - lim seq.|| + ||.lim seq.|| * |.(Cseq.n -
lim Cseq).| < (b * r + ||.lim seq.|| * r)/(b + ||.lim seq.||) by
XCMPLX_1:62;
then |.(Cseq.n).| * ||.seq.n - lim seq.|| + ||.lim seq.|| * |.(Cseq.n -
lim Cseq).| < ((b + ||.lim seq.||) * r)/(b + ||.lim seq.||);
then
A15: |.(Cseq.n).| * ||.seq.n - lim seq.|| + ||.lim seq.|| * |.(Cseq.n -
lim Cseq).| < r by A5,XCMPLX_1:89;
||.(Cseq * seq).n - lim Cseq * lim seq.|| = ||.Cseq.n * seq.n - lim
Cseq * lim seq.|| by Def8
.= ||.(Cseq.n * seq.n - lim Cseq * lim seq) + 09(X).|| by RLVECT_1:4
.= ||.(Cseq.n * seq.n - lim Cseq * lim seq) + (Cseq.n * lim seq - Cseq
.n * lim seq).|| by RLVECT_1:15
.= ||.Cseq.n * seq.n - (lim Cseq * lim seq - (Cseq.n * lim seq - Cseq.
n * lim seq)).|| by RLVECT_1:29
.= ||.Cseq.n * seq.n - (Cseq.n * lim seq + (lim Cseq * lim seq - Cseq.
n * lim seq)).|| by RLVECT_1:29
.= ||.(Cseq.n * seq.n - Cseq.n * lim seq) - (lim Cseq * lim seq - Cseq
.n * lim seq).|| by RLVECT_1:27
.= ||.(Cseq.n * seq.n - Cseq.n * lim seq) + (Cseq.n * lim seq - lim
Cseq * lim seq).|| by RLVECT_1:33;
then
||.(Cseq * seq).n - lim Cseq * lim seq.|| <= ||.Cseq.n * seq.n - Cseq
.n * lim seq.|| + ||.Cseq.n * lim seq - lim Cseq * lim seq.|| by CSSPACE:46
;
then
||.(Cseq * seq).n - lim Cseq * lim seq.|| <= ||.Cseq.n * (seq.n - lim
seq).|| + ||.Cseq.n * lim seq - lim Cseq * lim seq.|| by CLVECT_1:9;
then
||.(Cseq * seq).n - lim Cseq * lim seq.|| <= ||.Cseq.n * (seq.n - lim
seq).|| + ||.(Cseq.n - lim Cseq) * lim seq.|| by CLVECT_1:10;
then ||.(Cseq * seq).n - lim Cseq * lim seq.|| <= |.(Cseq.n).| * ||.seq.n
- lim seq.|| + ||.(Cseq.n - lim Cseq) * lim seq.|| by CSSPACE:43;
then ||.(Cseq * seq).n - lim Cseq * lim seq.|| <= |.(Cseq.n).| * ||.seq.n
- lim seq.|| + ||.lim seq.|| * |.(Cseq.n - lim Cseq).| by CSSPACE:43;
then ||.(Cseq * seq).n - lim Cseq * lim seq.|| < r by A15,XXREAL_0:2;
hence dist((Cseq * seq).n, (lim Cseq * lim seq)) < r by CSSPACE:def 16;
end;
Cseq * seq is convergent by A1,A2,Th48;
hence thesis by A7,CLVECT_2:def 2;
end;
definition
let Cseq;
attr Cseq is Cauchy means
for r st r > 0 ex k st for n, m st n >= k
& m >= k holds |.((Cseq.n - Cseq.m)).| < r;
end;
theorem
for X being ComplexHilbertSpace, seq being sequence of X
holds seq is Cauchy & Cseq is Cauchy implies Cseq *
seq is Cauchy
proof
let X be ComplexHilbertSpace, seq be sequence of X;
assume that
A1: seq is Cauchy and
A2: Cseq is Cauchy;
now
let r be Real;
assume r > 0;
then consider k such that
A3: for n, m st n >= k & m >= k holds |.((Cseq.n - Cseq.m)).| < r by A2;
take k;
thus for n st n >= k holds |.((Cseq.n - Cseq.k)).| < r by A3;
end;
then
A4: Cseq is convergent by COMSEQ_3:46;
seq is convergent by A1,CLVECT_2:def 11;
hence thesis by A4,Th48,CLVECT_2:65;
end;
theorem Th52:
for n holds Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n
= Partial_Sums(Cseq * seq).(n+1) - (Cseq * Partial_Sums(seq)).(n+1)
proof
defpred P[Nat] means
Partial_Sums((Cseq - Cseq^\1) * Partial_Sums
(seq)).$1 = Partial_Sums(Cseq * seq).($1+1) - (Cseq * Partial_Sums(seq)).($1+1)
;
A1: Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).0 = ((Cseq - Cseq^\1)
* Partial_Sums(seq)).0 by BHSP_4:def 1
.= (Cseq - Cseq^\1).0 * Partial_Sums(seq).0 by Def8
.= (Cseq + -Cseq^\1).0 * seq.0 by BHSP_4:def 1
.= (Cseq.0 + (-Cseq^\1).0) * seq.0 by VALUED_1:1
.= (Cseq.0 + -(Cseq^\1).0) * seq.0 by VALUED_1:8
.= (Cseq.0 - (Cseq^\1).0) * seq.0
.= (Cseq.0 - Cseq.(0+1)) * seq.0 by NAT_1:def 3
.= Cseq.0 * seq.0 - Cseq.1 * seq.0 by CLVECT_1:10;
A2: (Cseq * Partial_Sums(seq)).(0+1) = Cseq.(0+1) * Partial_Sums(seq).(0+1)
by Def8
.= Cseq.(0+1) * (Partial_Sums(seq).0 + seq.(0+1)) by BHSP_4:def 1
.= Cseq.1 * (seq.0 + seq.1) by BHSP_4:def 1
.= Cseq.1 * seq.0 + Cseq.1 * seq.1 by CLVECT_1:def 2;
A3: now
let n;
assume P[n];
then Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + (Cseq *
Partial_Sums(seq)).(n+1) = Partial_Sums(Cseq * seq).(n+1) - ((Cseq *
Partial_Sums(seq)).(n+1) - (Cseq * Partial_Sums(seq)).(n+1)) by RLVECT_1:29;
then
A4: Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + (Cseq *
Partial_Sums(seq)).(n+1) = Partial_Sums(Cseq * seq).(n+1) - 09(X) by
RLVECT_1:15;
Partial_Sums(Cseq * seq).((n+1)+1) = Partial_Sums(Cseq * seq).(n+1) +
(Cseq * seq).((n+1)+1) by BHSP_4:def 1
.= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + (Cseq *
Partial_Sums(seq)).(n+1) + (Cseq * seq).((n+1)+1) by A4,RLVECT_1:13
.= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + ((Cseq *
Partial_Sums(seq)).(n+1) + (Cseq * seq).((n+1)+1)) by RLVECT_1:def 3
.= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + (Cseq.(n+1)
* Partial_Sums(seq).(n+1) + (Cseq * seq).((n+1)+1)) by Def8
.= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + ( ((Cseq.(n+
1)-Cseq.((n+1)+1))+Cseq.((n+1)+1)) *Partial_Sums(seq).(n+1) + Cseq.((n+1)+1) *
seq.((n+1)+1) ) by Def8
.= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + ( ((Cseq.(n+
1) - Cseq.((n+1)+1)) * Partial_Sums(seq).(n+1) + Cseq.((n+1)+1) * Partial_Sums(
seq).(n+1)) + Cseq.((n+1)+1) * seq.((n+1)+1)) by CLVECT_1:def 3
.= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + (((Cseq.(n+1
) - (Cseq^\1).(n+1)) * Partial_Sums(seq).(n+1) + Cseq.((n+1)+1) * Partial_Sums(
seq).(n+1)) + Cseq.((n+1)+1) * seq.((n+1)+1)) by NAT_1:def 3
.= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + (((Cseq.(n+1
) + -(Cseq^\1).(n+1)) * Partial_Sums(seq).(n+1) + Cseq.((n+1)+1) * Partial_Sums
(seq).(n+1)) + Cseq.((n+1)+1) * seq.((n+1)+1))
.= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + (((Cseq.(n+1
) + (-Cseq^\1).(n+1)) * Partial_Sums(seq).(n+1) + Cseq.((n+1)+1) * Partial_Sums
(seq).(n+1)) + Cseq.((n+1)+1) * seq.((n+1)+1)) by VALUED_1:8
.= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + (((Cseq - (
Cseq^\1)).(n+1) * Partial_Sums(seq).(n+1) + Cseq.((n+1)+1) * Partial_Sums(seq).
(n+1)) + Cseq.((n+1)+1) * seq.((n+1)+1)) by VALUED_1:1
.= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + ((Cseq - (
Cseq^\1)).(n+1) * Partial_Sums(seq).(n+1) + (Cseq.((n+1)+1) * Partial_Sums(seq)
.(n+1) + Cseq.((n+1)+1) * seq.((n+1)+1))) by RLVECT_1:def 3
.= (Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + (Cseq - (
Cseq^\1)).(n+1) * Partial_Sums(seq).(n+1)) + (Cseq.((n+1)+1) * Partial_Sums(seq
).(n+1) + Cseq.((n+1)+1) * seq.((n+1)+1)) by RLVECT_1:def 3
.= (Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + ((Cseq -
Cseq^\1) * Partial_Sums(seq)).(n+1)) + (Cseq.((n+1)+1) * Partial_Sums(seq).(n+1
) + Cseq.((n+1)+1) * seq.((n+1)+1)) by Def8
.= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).(n+1) + (Cseq.((
n+1)+1) * Partial_Sums(seq).(n+1) + Cseq.((n+1)+1) * seq.((n+1)+1)) by
BHSP_4:def 1
.= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).(n+1) + (Cseq.((
n+1)+1) * (Partial_Sums(seq).(n+1) + seq.((n+1)+1))) by CLVECT_1:def 2
.= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).(n+1) + (Cseq.((
n+1)+1) * Partial_Sums(seq).((n+1)+1)) by BHSP_4:def 1
.= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).(n+1) + (Cseq *
Partial_Sums(seq)).((n+1)+1) by Def8;
then Partial_Sums(Cseq * seq).((n+1)+1) - (Cseq * Partial_Sums(seq)).((n+
1)+1) = Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).(n+1) + ((Cseq *
Partial_Sums(seq)).((n+1)+1) - (Cseq * Partial_Sums(seq)).((n+1)+1)) by
RLVECT_1:def 3;
then Partial_Sums(Cseq * seq).((n+1)+1) - (Cseq * Partial_Sums(seq)).((n+
1)+1) = Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).(n+1) + 09(X)
by RLVECT_1:15;
hence P[n+1] by RLVECT_1:4;
end;
Partial_Sums(Cseq * seq).(0+1) = Partial_Sums(Cseq * seq).0 + (Cseq *
seq).(0+1) by BHSP_4:def 1
.= (Cseq * seq).0 + (Cseq * seq).1 by BHSP_4:def 1
.= Cseq.0 * seq.0 + (Cseq * seq).1 by Def8
.= Cseq.0 * seq.0 + Cseq.1 * seq.1 by Def8;
then
Partial_Sums(Cseq * seq).(0+1) = (Cseq.0 * seq.0 + 09(X)) + Cseq.1 * seq
.1 by RLVECT_1:4
.= (Cseq.0 * seq.0 + (Cseq.1 * seq.0 - Cseq.1 * seq.0)) + Cseq.1 * seq.1
by RLVECT_1:15
.= ((Cseq.0 * seq.0 + -(Cseq.1 * seq.0)) + Cseq.1 * seq.0) + Cseq.1 *
seq.1 by RLVECT_1:def 3
.= Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).0 + (Cseq *
Partial_Sums(seq)).(0+1) by A1,A2,RLVECT_1:def 3;
then Partial_Sums(Cseq * seq).(0+1) - (Cseq * Partial_Sums(seq)).(0+1) =
Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).0 + ((Cseq * Partial_Sums(
seq)).(0+1) - (Cseq * Partial_Sums(seq)).(0+1)) by RLVECT_1:def 3;
then Partial_Sums(Cseq * seq).(0+1) - (Cseq * Partial_Sums(seq)).(0+1) =
Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).0 + 09(X) by RLVECT_1:15;
then
A5: P[0] by RLVECT_1:4;
thus for n holds P[n] from NAT_1:sch 2(A5,A3);
end;
theorem Th53:
for n holds Partial_Sums(Cseq * seq).(n+1) = (Cseq *
Partial_Sums(seq)).(n+1) - Partial_Sums((Cseq^\1 - Cseq) * Partial_Sums(seq)).n
proof
let n;
Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + (Cseq *
Partial_Sums(seq)).(n+1) = (Partial_Sums(Cseq * seq).(n+1) - (Cseq *
Partial_Sums(seq)).(n+1)) + (Cseq * Partial_Sums(seq)).(n+1) by Th52;
then Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + (Cseq *
Partial_Sums(seq)).(n+1) = Partial_Sums(Cseq * seq).(n+1) - ((Cseq *
Partial_Sums(seq)).(n+1) - (Cseq * Partial_Sums(seq)).(n+1)) by RLVECT_1:29;
then Partial_Sums((Cseq - Cseq^\1) * Partial_Sums(seq)).n + (Cseq *
Partial_Sums(seq)).(n+1) = Partial_Sums(Cseq * seq).(n+1) - 09(X) by
RLVECT_1:15;
then Partial_Sums(Cseq * seq).(n+1) = (Cseq * Partial_Sums(seq)).(n+1) +
Partial_Sums((Cseq + -(Cseq^\1))*Partial_Sums(seq)).n by RLVECT_1:13;
then Partial_Sums(Cseq * seq).(n+1) = (Cseq * Partial_Sums(seq)).(n+1) +
Partial_Sums(((-1r) (#) (Cseq^\1) - -Cseq) * Partial_Sums(seq)).n by
COMSEQ_1:11;
then Partial_Sums(Cseq * seq).(n+1) = (Cseq * Partial_Sums(seq)).(n+1) +
Partial_Sums(((-1r) (#) (Cseq^\1) - (-1r) (#) Cseq) * Partial_Sums(seq)).n
by COMSEQ_1:11;
then Partial_Sums(Cseq * seq).(n+1) = (Cseq * Partial_Sums(seq)).(n+1) +
Partial_Sums(((-1r) (#) (Cseq^\1 - Cseq)) * Partial_Sums(seq)).n by
COMSEQ_1:18;
then Partial_Sums(Cseq * seq).(n+1) = (Cseq * Partial_Sums(seq)).(n+1) +
Partial_Sums((-1r) * ((Cseq^\1 - Cseq) * Partial_Sums(seq))).n by Th46;
then
Partial_Sums(Cseq * seq).(n+1) = (Cseq * Partial_Sums(seq)).(n+1) + ((-
1r) * Partial_Sums((Cseq^\1 - Cseq) * Partial_Sums(seq))).n by Th3;
then Partial_Sums(Cseq * seq).(n+1) = (Cseq * Partial_Sums(seq)).(n+1) + (-
1r) * Partial_Sums((Cseq^\1 - Cseq) * Partial_Sums(seq)).n by CLVECT_1:def 14
;
hence thesis by CLVECT_1:3;
end;
theorem
for n holds Sum(Cseq*seq,n+1) = (Cseq*Partial_Sums(seq)).(n+1) - Sum((
Cseq^\1 - Cseq)*Partial_Sums(seq),n) by Th53;