Volume 3, 1991

University of Bialystok

Copyright (c) 1991 Association of Mizar Users

### The abstract of the Mizar article:

### Introduction to Banach and Hilbert Spaces --- Part III

**by****Jan Popiolek**- Received July 19, 1991
- MML identifier: BHSP_3

- [ Mizar article, MML identifier index ]

environ vocabulary BHSP_1, PRE_TOPC, NORMSP_1, ORDINAL2, SEQM_3, SEQ_1, RLVECT_1, METRIC_1, FUNCT_1, ARYTM_1, ARYTM_3, ABSVALUE, RELAT_1, SEQ_2, LATTICES, BHSP_3, ARYTM; notation TARSKI, ORDINAL1, NUMBERS, XREAL_0, REAL_1, NAT_1, RELAT_1, FUNCT_1, FUNCT_2, SEQ_1, SEQM_3, ABSVALUE, STRUCT_0, PRE_TOPC, RLVECT_1, NORMSP_1, BHSP_1, BHSP_2; constructors REAL_1, NAT_1, SEQ_1, ABSVALUE, BHSP_2, MEMBERED, XBOOLE_0; clusters FUNCT_1, SEQM_3, STRUCT_0, XREAL_0, RELSET_1, ARYTM_3, MEMBERED, ZFMISC_1, XBOOLE_0, NUMBERS, ORDINAL2; requirements REAL, NUMERALS, SUBSET, BOOLE, 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,Nseq1,Nseq2 for increasing Seq_of_Nat, Rseq for Real_Sequence, k, l, l1, l2, l3, n, m, m1, m2 for Nat; 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; synonym seq is_Cauchy_sequence; end; theorem :: BHSP_3:1 seq is constant implies seq is_Cauchy_sequence; theorem :: BHSP_3:2 seq is_Cauchy_sequence iff for r st r > 0 ex k st for n, m st ( n >= k & m >= k ) holds ||.(seq.n) - (seq.m).|| < r; theorem :: BHSP_3:3 seq1 is_Cauchy_sequence & seq2 is_Cauchy_sequence implies seq1 + seq2 is_Cauchy_sequence; theorem :: BHSP_3:4 seq1 is_Cauchy_sequence & seq2 is_Cauchy_sequence implies seq1 - seq2 is_Cauchy_sequence; theorem :: BHSP_3:5 seq is_Cauchy_sequence implies a * seq is_Cauchy_sequence; theorem :: BHSP_3:6 seq is_Cauchy_sequence implies - seq is_Cauchy_sequence; theorem :: BHSP_3:7 seq is_Cauchy_sequence implies seq + x is_Cauchy_sequence; theorem :: BHSP_3:8 seq is_Cauchy_sequence implies seq - x is_Cauchy_sequence; theorem :: BHSP_3:9 seq is convergent implies seq is_Cauchy_sequence; 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; end; theorem :: BHSP_3:10 seq is_compared_to seq; theorem :: BHSP_3:11 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 :: 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_sequence & seq1 is_compared_to seq2 implies seq2 is_Cauchy_sequence; 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; theorem :: BHSP_3:18 seq1 is bounded & seq2 is bounded implies seq1 + seq2 is bounded; theorem :: BHSP_3:19 seq is bounded implies -seq is bounded; theorem :: BHSP_3:20 seq1 is bounded & seq2 is bounded implies seq1 - seq2 is bounded; theorem :: BHSP_3:21 seq is bounded implies a * seq is bounded; theorem :: BHSP_3:22 seq is constant implies seq is bounded; theorem :: BHSP_3:23 for m ex M st ( M > 0 & for n st n <= m holds ||.seq.n.|| < M ); theorem :: BHSP_3:24 seq is convergent implies seq is bounded; theorem :: BHSP_3:25 seq1 is bounded & seq1 is_compared_to seq2 implies seq2 is bounded; definition let X, Nseq, seq; redefine func seq * Nseq -> sequence of X; end; definition let X be non empty 1-sorted, s1, s be sequence of X; pred s1 is_subsequence_of s means :: BHSP_3:def 4 ex N being increasing Seq_of_Nat st s1 = s * N; end; theorem :: BHSP_3:26 for X being RealUnitarySpace, s being sequence of X, N being increasing Seq_of_Nat for n being Nat holds (s * N).n=s.(N.n); theorem :: BHSP_3:27 seq is_subsequence_of seq; theorem :: BHSP_3:28 seq1 is_subsequence_of seq2 & seq2 is_subsequence_of seq3 implies seq1 is_subsequence_of seq3; theorem :: BHSP_3:29 seq is constant & seq1 is_subsequence_of seq implies seq1 is constant; theorem :: BHSP_3:30 seq is constant & seq1 is_subsequence_of seq implies seq = seq1; theorem :: BHSP_3:31 seq is bounded & seq1 is_subsequence_of seq implies seq1 is bounded; theorem :: BHSP_3:32 seq is convergent & seq1 is_subsequence_of seq implies seq1 is convergent; theorem :: BHSP_3:33 seq1 is_subsequence_of seq & seq is convergent implies lim seq1=lim seq; theorem :: BHSP_3:34 seq is_Cauchy_sequence & seq1 is_subsequence_of seq implies seq1 is_Cauchy_sequence; definition let X; let seq; let k; func seq ^\k -> sequence of X means :: BHSP_3:def 5 for n holds it.n=seq.(n + k); end; theorem :: BHSP_3:35 seq ^\0 = seq; theorem :: BHSP_3:36 (seq ^\k)^\m = (seq ^\m)^\k; theorem :: BHSP_3:37 (seq ^\k)^\m=seq ^\(k + m); theorem :: BHSP_3:38 (seq1 + seq2) ^\k = (seq1 ^\k) + (seq2 ^\k); theorem :: BHSP_3:39 (-seq) ^\k = -(seq ^\k); theorem :: BHSP_3:40 (seq1 - seq2) ^\k = (seq1 ^\k) - (seq2 ^\k); theorem :: BHSP_3:41 (a * seq) ^\k = a * (seq ^\k); theorem :: BHSP_3:42 (seq * Nseq) ^\k = seq * (Nseq ^\k); theorem :: BHSP_3:43 seq ^\k is_subsequence_of seq; theorem :: BHSP_3:44 seq is convergent implies ((seq ^\k) is convergent & lim (seq ^\k)=lim seq); canceled; theorem :: BHSP_3:46 seq is convergent & (ex k st seq = seq1 ^\k) implies seq1 is convergent; theorem :: BHSP_3:47 seq is_Cauchy_sequence & (ex k st seq = seq1 ^\k) implies seq1 is_Cauchy_sequence; theorem :: BHSP_3:48 seq is_Cauchy_sequence implies (seq ^\k) is_Cauchy_sequence; theorem :: BHSP_3:49 seq1 is_compared_to seq2 implies (seq1 ^\k) is_compared_to (seq2 ^\k); theorem :: BHSP_3:50 seq is bounded implies (seq ^\k) is bounded; theorem :: BHSP_3:51 seq is constant implies (seq ^\k) is constant; definition let X; attr X is complete means :: BHSP_3:def 6 for seq holds seq is_Cauchy_sequence implies seq is convergent; synonym X is_complete_space; end; canceled; theorem :: BHSP_3:53 X is_complete_space & seq is_Cauchy_sequence implies seq is bounded; definition let X; attr X is Hilbert means :: BHSP_3:def 7 X is RealUnitarySpace & X is_complete_space; synonym X is_Hilbert_space; end;

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