Volume 4, 1992

University of Bialystok

Copyright (c) 1992 Association of Mizar Users

### The abstract of the Mizar article:

### Series in Banach and Hilbert Spaces

**by****Elzbieta Kraszewska, and****Jan Popiolek**- Received April 1, 1992
- MML identifier: BHSP_4

- [ Mizar article, MML identifier index ]

environ vocabulary BHSP_1, PRE_TOPC, NORMSP_1, SEQ_1, RLVECT_1, FUNCT_1, SERIES_1, ARYTM_1, SUPINF_2, SEQ_2, ORDINAL2, SEQM_3, BHSP_3, LATTICES, ABSVALUE, ARYTM_3, POWER, PROB_1, INT_1, METRIC_1, ARYTM, GROUP_1; notation SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, STRUCT_0, REAL_1, NAT_1, FUNCT_2, SEQ_1, SEQ_2, SEQM_3, SERIES_1, ABSVALUE, PRE_TOPC, RLVECT_1, NORMSP_1, BHSP_1, BHSP_2, BHSP_3, PREPOWER, POWER, INT_1; constructors REAL_1, NAT_1, SEQ_2, SERIES_1, BHSP_2, BHSP_3, PREPOWER, PARTFUN1, MEMBERED, XBOOLE_0; clusters INT_1, STRUCT_0, XREAL_0, SEQ_1, RELSET_1, MEMBERED, ZFMISC_1, XBOOLE_0, ORDINAL2; requirements NUMERALS, REAL, SUBSET, ARITHM; begin reserve X for RealUnitarySpace; reserve g for Point of X; reserve a, b, r, M2 for Real; reserve seq, seq1, seq2 for sequence of X; reserve Rseq,Rseq1,Rseq2 for Real_Sequence; reserve k, n, m, m1, m2 for Nat; scheme Rec_Func_Ex_RUS{ X() -> RealUnitarySpace, z() -> Point of X(), G(Nat, Point of X()) -> Point of X()}: ex f being Function of NAT, the carrier of X() st f.0 = z() & for n being Element of NAT, x being Point of X() st x = f.n holds f.(n + 1) = G(n,x); definition let X; let seq; func Partial_Sums(seq) -> sequence of X means :: BHSP_4:def 1 it.0 = seq.0 & for n holds it.(n + 1) = it.n + seq.(n + 1); end; theorem :: BHSP_4:1 Partial_Sums(seq1) + Partial_Sums(seq2) = Partial_Sums(seq1 + seq2); theorem :: BHSP_4:2 Partial_Sums(seq1) - Partial_Sums(seq2) = Partial_Sums(seq1 - seq2); theorem :: BHSP_4:3 Partial_Sums(a * seq) = a * Partial_Sums(seq); theorem :: BHSP_4:4 Partial_Sums(- seq) = - Partial_Sums(seq); theorem :: BHSP_4:5 a * Partial_Sums(seq1) + b * Partial_Sums(seq2) = Partial_Sums(a * seq1 + b * seq2); definition let X; let seq; attr seq is summable means :: BHSP_4:def 2 Partial_Sums(seq) is convergent; func Sum(seq) -> Point of X equals :: BHSP_4:def 3 lim Partial_Sums(seq); end; theorem :: BHSP_4:6 seq1 is summable & seq2 is summable implies seq1 + seq2 is summable & Sum(seq1 + seq2) = Sum(seq1) + Sum(seq2); theorem :: BHSP_4:7 seq1 is summable & seq2 is summable implies seq1 - seq2 is summable & Sum(seq1 - seq2) = Sum(seq1) - Sum(seq2); theorem :: BHSP_4:8 seq is summable implies a * seq is summable & Sum(a * seq) = a * Sum(seq); theorem :: BHSP_4:9 seq is summable implies seq is convergent & lim seq = 0.X; theorem :: BHSP_4:10 X is_Hilbert_space implies ( 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 )); theorem :: BHSP_4:11 seq is summable implies Partial_Sums(seq) is bounded; theorem :: BHSP_4:12 for seq, seq1 st for n holds seq1.n = seq.0 holds Partial_Sums(seq^\1) = (Partial_Sums(seq)^\1) - seq1; theorem :: BHSP_4:13 seq is summable implies for k holds seq^\k is summable; theorem :: BHSP_4:14 (ex k st seq^\k is summable) implies seq is summable; definition let X, seq, n; func Sum(seq,n) -> Point of X equals :: BHSP_4:def 4 Partial_Sums(seq).n; end; canceled; theorem :: BHSP_4:16 Sum(seq, 0) = seq.0; theorem :: BHSP_4:17 Sum(seq, 1) = Sum(seq, 0) + seq.1; theorem :: BHSP_4:18 Sum(seq, 1) = seq.0 + seq.1; theorem :: BHSP_4:19 Sum(seq, n + 1) = Sum(seq, n) + seq.(n + 1); theorem :: BHSP_4:20 seq.(n + 1) = Sum(seq, n + 1) - Sum(seq, n); theorem :: BHSP_4:21 seq.1 = Sum(seq, 1) - Sum(seq, 0); definition let X, seq, n, m; func Sum(seq, n, m) -> Point of X equals :: BHSP_4:def 5 Sum(seq, n) - Sum(seq, m); end; canceled; theorem :: BHSP_4:23 Sum(seq, 1, 0) = seq.1; theorem :: BHSP_4:24 Sum(seq, n+1, n) = seq.(n+1); theorem :: BHSP_4:25 X is_Hilbert_space implies ( 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 )); theorem :: BHSP_4:26 X is_Hilbert_space implies ( 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 )); definition let Rseq, n; func Sum(Rseq, n) -> Real equals :: BHSP_4:def 6 Partial_Sums(Rseq).n; end; definition let Rseq, n,m; func Sum(Rseq, n, m) -> Real equals :: BHSP_4:def 7 Sum(Rseq, n) - Sum(Rseq, m); end; definition let X, seq; attr seq is absolutely_summable means :: BHSP_4:def 8 ||.seq.|| is summable; end; theorem :: BHSP_4:27 seq1 is absolutely_summable & seq2 is absolutely_summable implies seq1 + seq2 is absolutely_summable; theorem :: BHSP_4:28 seq is absolutely_summable implies a * seq is absolutely_summable; theorem :: BHSP_4:29 ( for n holds ||.seq.||.n <= Rseq.n ) & Rseq is summable implies seq is absolutely_summable; theorem :: BHSP_4:30 ( 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; theorem :: BHSP_4:31 r > 0 & ( ex m st for n st n >= m holds ||.seq.n.|| >= r) implies not seq is convergent or lim seq <> 0.X; theorem :: BHSP_4:32 ( 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; theorem :: BHSP_4:33 (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; theorem :: BHSP_4:34 ( for n holds Rseq.n = n-root (||.seq.n.||) ) & Rseq is convergent & lim Rseq < 1 implies seq is absolutely_summable; theorem :: BHSP_4:35 (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; theorem :: BHSP_4:36 (for n holds Rseq.n = n-root (||.seq.||.n)) & Rseq is convergent & lim Rseq > 1 implies not seq is summable; theorem :: BHSP_4:37 Partial_Sums(||.seq.||) is non-decreasing; theorem :: BHSP_4:38 for n holds Partial_Sums(||.seq.||).n >= 0; theorem :: BHSP_4:39 for n holds ||.Partial_Sums(seq).n.|| <= Partial_Sums(||.seq.||).n; theorem :: BHSP_4:40 for n holds ||.Sum(seq, n).|| <= Sum(||.seq.||, n); theorem :: BHSP_4:41 for n, m holds ||.Partial_Sums(seq).m - Partial_Sums(seq).n.|| <= abs(Partial_Sums(||.seq.||).m - Partial_Sums(||.seq.||).n); theorem :: BHSP_4:42 for n, m holds ||.Sum(seq, m) - Sum(seq, n).|| <= abs(Sum(||.seq.||, m) - Sum(||.seq.||, n)); theorem :: BHSP_4:43 for n, m holds ||.Sum(seq, m, n).|| <= abs(Sum(||.seq.||, m, n)); theorem :: BHSP_4:44 X is_Hilbert_space implies ( seq is absolutely_summable implies seq is summable ); definition let X, seq, Rseq; func Rseq * seq -> sequence of X means :: BHSP_4:def 9 for n holds it.n = Rseq.n * seq.n; end; theorem :: BHSP_4:45 Rseq * (seq1 + seq2) = Rseq * seq1 + Rseq * seq2; theorem :: BHSP_4:46 (Rseq1 + Rseq2) * seq = Rseq1 * seq + Rseq2 * seq; theorem :: BHSP_4:47 (Rseq1 (#) Rseq2) * seq = Rseq1 * (Rseq2 * seq); theorem :: BHSP_4:48 (a (#) Rseq) * seq = a * (Rseq * seq); theorem :: BHSP_4:49 Rseq * (- seq) = (- Rseq) * seq; theorem :: BHSP_4:50 Rseq is convergent & seq is convergent implies Rseq * seq is convergent; theorem :: BHSP_4:51 Rseq is bounded & seq is bounded implies Rseq * seq is bounded; theorem :: BHSP_4:52 Rseq is convergent & seq is convergent implies Rseq * seq is convergent & lim (Rseq * seq) = lim Rseq * lim seq; definition let Rseq; attr Rseq is Cauchy means :: BHSP_4:def 10 for r st r > 0 ex k st for n, m st ( n >= k & m >= k ) holds abs((Rseq.n - Rseq.m)) < r; synonym Rseq is_Cauchy_sequence; end; theorem :: BHSP_4:53 X is_Hilbert_space implies ( seq is_Cauchy_sequence & Rseq is_Cauchy_sequence implies Rseq * seq is_Cauchy_sequence ); theorem :: BHSP_4:54 for n holds Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n = Partial_Sums(Rseq * seq).(n+1) - (Rseq * Partial_Sums(seq)).(n+1); theorem :: BHSP_4:55 for n holds Partial_Sums(Rseq * seq).(n+1) = (Rseq * Partial_Sums(seq)).(n+1) - Partial_Sums((Rseq^\1 - Rseq) * Partial_Sums(seq)).n; theorem :: BHSP_4:56 for n holds Sum(Rseq * seq, n+1) = (Rseq * Partial_Sums(seq)).(n+1) - Sum((Rseq^\1 - Rseq) * Partial_Sums(seq), n);

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