Volume 15, 2003

University of Bialystok

Copyright (c) 2003 Association of Mizar Users

### The abstract of the Mizar article:

### Hilbert Space of Real Sequences

**by****Noboru Endou,****Yasumasa Suzuki, and****Yasunari Shidama**- Received April 3, 2003
- MML identifier: RSSPACE2

- [ Mizar article, MML identifier index ]

environ vocabulary PRE_TOPC, NORMSP_1, RLVECT_1, FUNCT_1, ARYTM, ARYTM_1, ARYTM_3, RELAT_1, ABSVALUE, ORDINAL2, SQUARE_1, PROB_1, SEQ_1, SEQ_2, SEQM_3, SERIES_1, PROB_2, BHSP_1, BHSP_3, SUPINF_2, RSSPACE; notation TARSKI, SUBSET_1, ZFMISC_1, ORDINAL1, NUMBERS, XCMPLX_0, STRUCT_0, XREAL_0, REAL_1, NAT_1, RELAT_1, PARTFUN1, FUNCT_1, FUNCT_2, PRE_TOPC, RLVECT_1, ABSVALUE, NORMSP_1, BHSP_1, BHSP_2, BHSP_3, SQUARE_1, SEQ_1, SEQ_2, SEQM_3, SERIES_1, RSSPACE; constructors REAL_1, NAT_1, DOMAIN_1, SQUARE_1, SEQ_2, SERIES_1, PREPOWER, PARTFUN1, RLSUB_1, BHSP_2, BHSP_3, RSSPACE, MEMBERED; clusters RELSET_1, XREAL_0, STRUCT_0, SEQ_1, RSSPACE, MEMBERED, NUMBERS, ORDINAL2; requirements SUBSET, REAL, BOOLE, NUMERALS, ARITHM; begin :: Hilbert Space of Real Sequences theorem :: RSSPACE2:1 ( the carrier of l2_Space = the_set_of_l2RealSequences) & (for x be set holds x is Element of l2_Space iff x is Real_Sequence & seq_id(x)(#)seq_id(x) is summable) & (for x be set holds x is VECTOR of l2_Space iff x is Real_Sequence & seq_id(x)(#)seq_id(x) is summable) & 0.l2_Space = Zeroseq & (for u be VECTOR of l2_Space holds u =seq_id(u)) & (for u,v be VECTOR of l2_Space holds u+v =seq_id(u)+seq_id(v)) & (for r be Real for u be VECTOR of l2_Space holds r*u =r(#)seq_id(u)) & (for u be VECTOR of l2_Space holds -u =-seq_id(u) & seq_id(-u)=-seq_id(u)) & (for u,v be VECTOR of l2_Space holds u-v =seq_id(u)-seq_id(v)) & for v,w be VECTOR of l2_Space holds seq_id(v)(#)seq_id(w) is summable & for v,w be VECTOR of l2_Space holds v.|.w = Sum(seq_id(v)(#)seq_id(w)); theorem :: RSSPACE2:2 for x, y, z being Point of l2_Space for a be Real holds ( x .|. x = 0 iff x = 0.l2_Space ) & 0 <= x .|. x & x .|. y = y .|. x & (x+y) .|. z = x .|. z + y .|. z & (a*x) .|. y = a * ( x .|. y ); definition cluster l2_Space -> RealUnitarySpace-like; end; theorem :: RSSPACE2:3 for vseq be sequence of l2_Space st vseq is_Cauchy_sequence holds vseq is convergent; definition cluster l2_Space -> Hilbert complete; end; begin :: Miscellaneous theorem :: RSSPACE2:4 for rseq be Real_Sequence st ((for n be Nat holds 0 <= rseq.n) & rseq is summable) holds ( for n be Nat holds rseq.n <= (Partial_Sums rseq).n ) & ( for n be Nat holds 0 <= (Partial_Sums rseq).n ) & ( for n be Nat holds (Partial_Sums(rseq)).n <= Sum rseq ) & ( for n be Nat holds rseq.n <= Sum(rseq) ); theorem :: RSSPACE2:5 (for x,y be Real holds (x+y)*(x+y) <= 2*x*x + 2*y*y) & (for x,y be Real holds x*x <= 2*(x-y)*(x-y) + 2*y*y); theorem :: RSSPACE2:6 for e being Real, seq being Real_Sequence st (seq is convergent & ex k being Nat st (for i being Nat st k <= i holds seq.i <=e)) holds lim seq <=e; theorem :: RSSPACE2:7 for c being Real, seq being Real_Sequence st seq is convergent for rseq be Real_Sequence st (for i be Nat holds rseq.i = (seq.i-c)*(seq.i-c)) holds rseq is convergent & lim rseq = (lim seq-c)*(lim seq-c); theorem :: RSSPACE2:8 for c being Real, seq,seq1 being Real_Sequence st seq is convergent & seq1 is convergent for rseq be Real_Sequence st (for i be Nat holds rseq.i = (seq.i-c)*(seq.i-c)+seq1.i) holds rseq is convergent & lim rseq = (lim seq-c)*(lim seq-c)+lim seq1;

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