Volume 15, 2003

University of Bialystok

Copyright (c) 2003 Association of Mizar Users

### The abstract of the Mizar article:

### On Some Properties of Real Hilbert Space. Part I

**by****Hiroshi Yamazaki,****Yasumasa Suzuki,****Takao Inoue, and****Yasunari Shidama**- Received February 25, 2003
- MML identifier: BHSP_6

- [ Mizar article, MML identifier index ]

environ vocabulary BOOLE, PRE_TOPC, NORMSP_1, SEQ_2, RLVECT_1, FUNCT_1, ARYTM_1, FINSET_1, BHSP_1, BHSP_3, ABSVALUE, RELAT_1, ARYTM_3, BINOP_1, FUNCT_2, VECTSP_1, HAHNBAN, FINSEQ_1, SETWISEO, CARD_1, FINSOP_1, BHSP_5, BHSP_6, SEMI_AF1, JORDAN2C; notation TARSKI, SUBSET_1, XBOOLE_0, ZFMISC_1, CARD_1, NUMBERS, XREAL_0, REAL_1, NAT_1, RELAT_1, FUNCT_1, FUNCT_2, FINSET_1, ABSVALUE, STRUCT_0, PRE_TOPC, RLVECT_1, BHSP_1, BHSP_2, BHSP_3, NORMSP_1, BINOP_1, VECTSP_1, SETWISEO, HAHNBAN, FINSEQ_1, FINSOP_1, BHSP_5; constructors REAL_1, NAT_1, BHSP_2, BHSP_3, PREPOWER, SETWISEO, VECTSP_1, HAHNBAN, BINOP_1, FINSOP_1, BHSP_5, MEMBERED; clusters RELSET_1, FUNCT_1, SUBSET_1, FINSET_1, STRUCT_0, NAT_1, XREAL_0, MEMBERED, NUMBERS, ORDINAL2; requirements SUBSET, REAL, BOOLE, NUMERALS, ARITHM; begin :: Preliminaries reserve X for RealUnitarySpace; reserve x for Point of X; reserve i, n for Nat; definition let X such that the add of X is commutative associative & the add of X has_a_unity; let Y be finite Subset of X; func setsum Y -> Element of X means :: BHSP_6:def 1 ex p being FinSequence of the carrier of X st p is one-to-one & rng p = Y & it = (the add of X) "**" p; end; theorem :: BHSP_6:1 for X st the add of X is commutative associative & the add of X has_a_unity for Y be finite Subset of X for I be Function of the carrier of X,the carrier of X st Y c= dom I & for x be set st x in dom I holds I.x = x holds setsum(Y) = setopfunc(Y, the carrier of X, the carrier of X, I, the add of X); theorem :: BHSP_6:2 for X st the add of X is commutative associative & the add of X has_a_unity for Y1, Y2 be finite Subset of X st Y1 misses Y2 for Z be finite Subset of X st Z = Y1 \/ Y2 holds setsum(Z) = setsum(Y1) + setsum(Y2); begin :: Summability definition let X; let Y be Subset of X; attr Y is summable_set means :: BHSP_6:def 2 ex x st for e be Real st e > 0 ex Y0 be finite Subset of X st Y0 is non empty & Y0 c= Y & for Y1 be finite Subset of X st Y0 c= Y1 & Y1 c= Y holds ||.x-setsum(Y1).|| < e; end; definition let X; let Y be Subset of X; assume Y is summable_set; func sum Y -> Point of X means :: BHSP_6:def 3 for e be Real st e > 0 ex Y0 be finite Subset of X st Y0 is non empty & Y0 c= Y & for Y1 be finite Subset of X st Y0 c= Y1 & Y1 c= Y holds ||.it-setsum(Y1).|| < e; end; definition let X; let L be linear-Functional of X; attr L is Bounded means :: BHSP_6:def 4 ex K be Real st K > 0 & for x holds abs(L.x) <= K * ||.x.||; end; definition let X; let Y be Subset of X; attr Y is weakly_summable_set means :: BHSP_6:def 5 ex x st for L be linear-Functional of X st L is Bounded for e be Real st e > 0 ex Y0 be finite Subset of X st Y0 is non empty & Y0 c= Y & for Y1 be finite Subset of X st Y0 c= Y1 & Y1 c= Y holds abs(L.(x-setsum(Y1))) < e; end; definition let X; let Y be Subset of X; let L be Functional of X; pred Y is_summable_set_by L means :: BHSP_6:def 6 ex r be Real st for e be Real st e > 0 ex Y0 be finite Subset of X st Y0 is non empty & Y0 c= Y & for Y1 be finite Subset of X st Y0 c= Y1 & Y1 c= Y holds abs(r-setopfunc(Y1,the carrier of X,REAL, L, addreal)) < e; end; definition let X; let Y be Subset of X; let L be Functional of X; assume Y is_summable_set_by L; func sum_byfunc(Y,L) -> Real means :: BHSP_6:def 7 for e be Real st e > 0 ex Y0 be finite Subset of X st Y0 is non empty & Y0 c= Y & for Y1 be finite Subset of X st Y0 c= Y1 & Y1 c= Y holds abs(it-setopfunc(Y1,the carrier of X,REAL, L, addreal)) < e; end; theorem :: BHSP_6:3 for Y be Subset of X st Y is summable_set holds Y is weakly_summable_set; theorem :: BHSP_6:4 for L be linear-Functional of X for p be FinSequence of the carrier of X st len p >=1 for q be FinSequence of REAL st dom p = dom q & (for i st i in dom q holds q.i = L.(p.i) ) holds L.((the add of X) "**" p) = addreal "**" q; theorem :: BHSP_6:5 for X st the add of X is commutative associative & the add of X has_a_unity for S be finite Subset of X st S is non empty for L be linear-Functional of X holds L.(setsum(S)) = setopfunc(S,the carrier of X,REAL, L, addreal); theorem :: BHSP_6:6 for X st the add of X is commutative associative & the add of X has_a_unity for Y be Subset of X holds Y is weakly_summable_set implies ex x st for L be linear-Functional of X st L is Bounded for e be Real st e > 0 ex Y0 be finite Subset of X st Y0 is non empty & Y0 c= Y & for Y1 be finite Subset of X st Y0 c= Y1 & Y1 c= Y holds abs((L.x)-(setopfunc(Y1,the carrier of X,REAL, L, addreal))) < e; theorem :: BHSP_6:7 for X st the add of X is commutative associative & the add of X has_a_unity for Y be Subset of X st Y is weakly_summable_set for L be linear-Functional of X st L is Bounded holds Y is_summable_set_by L; theorem :: BHSP_6:8 for X st the add of X is commutative associative & the add of X has_a_unity for Y be Subset of X st Y is summable_set for L be linear-Functional of X st L is Bounded holds Y is_summable_set_by L; theorem :: BHSP_6:9 for Y be finite Subset of X st Y is non empty holds Y is summable_set; begin :: Necessary and sufficient condition for summability theorem :: BHSP_6:10 for X st the add of X is commutative associative & the add of X has_a_unity & X is_Hilbert_space for Y be Subset of X holds Y is summable_set iff (for e be Real st e > 0 holds ex Y0 be finite Subset of X st Y0 is non empty & Y0 c= Y & (for Y1 be finite Subset of X st Y1 is non empty & Y1 c= Y & Y0 misses Y1 holds ||.setsum(Y1).|| < e));

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