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 II

**by****Hiroshi Yamazaki,****Yasumasa Suzuki,****Takao Inoue, and****Yasunari Shidama**- Received April 17, 2003
- MML identifier: BHSP_7

- [ Mizar article, MML identifier index ]

environ vocabulary BOOLE, PRE_TOPC, NORMSP_1, RLVECT_1, FUNCT_1, ARYTM, ARYTM_1, ARYTM_3, RELAT_1, ABSVALUE, BINOP_1, SQUARE_1, SEQ_2, PROB_2, BHSP_1, BHSP_3, FINSET_1, VECTSP_1, HAHNBAN, FINSEQ_1, SETWISEO, FINSOP_1, BHSP_5, BHSP_6, SEMI_AF1; notation TARSKI, SUBSET_1, XBOOLE_0, ZFMISC_1, ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, STRUCT_0, REAL_1, NAT_1, FUNCT_1, FUNCT_2, RELAT_1, PRE_TOPC, RLVECT_1, ABSVALUE, BHSP_1, BHSP_3, SQUARE_1, SEQ_2, BINOP_1, FINSET_1, VECTSP_1, SETWISEO, HAHNBAN, FINSEQ_1, FINSOP_1, BHSP_5, BHSP_6; constructors REAL_1, NAT_1, DOMAIN_1, SQUARE_1, SEQ_2, PREPOWER, BINOP_1, BHSP_3, SETWISEO, HAHNBAN1, FINSOP_1, BHSP_5, BHSP_6, MEMBERED; clusters RELSET_1, STRUCT_0, XREAL_0, XBOOLE_0, FINSET_1, BHSP_5, MEMBERED, NUMBERS, ORDINAL2; requirements SUBSET, REAL, BOOLE, NUMERALS, ARITHM; begin :: Necessary and sufficient condition for summable set reserve X for RealUnitarySpace; reserve x, y, y1, y2 for Point of X; theorem :: BHSP_7:1 for Y be Subset of X for L be Functional of X holds Y is_summable_set_by L iff (for e be Real st 0 < e 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 abs(setopfunc(Y1, the carrier of X, REAL, L, addreal)) <e )); theorem :: BHSP_7:2 for X st the add of X is commutative associative & the add of X has_a_unity for S be finite OrthogonalFamily of X st S is non empty for I be Function of the carrier of X, the carrier of X st S c= dom I & (for y st y in S holds I.y = y) for H be Function of the carrier of X, REAL st S c= dom H & (for y st y in S holds H.y= (y.|.y)) holds setopfunc(S, the carrier of X, the carrier of X, I, the add of X) .|. setopfunc(S, the carrier of X, the carrier of X, I, the add of X) = setopfunc(S, the carrier of X, REAL, H, addreal); theorem :: BHSP_7:3 for X st the add of X is commutative associative & the add of X has_a_unity for S be finite OrthogonalFamily of X st S is non empty for H be Functional of X st S c= dom H & (for x st x in S holds H.x = (x.|.x)) holds (setsum(S)).|.(setsum(S)) = setopfunc(S, the carrier of X, REAL, H, addreal); theorem :: BHSP_7:4 for Y be OrthogonalFamily of X for Z be Subset of X holds Z is Subset of Y implies Z is OrthogonalFamily of X; theorem :: BHSP_7:5 for Y be OrthonormalFamily of X for Z be Subset of X holds Z is Subset of Y implies Z is OrthonormalFamily of X; begin :: Equivalence of summability theorem :: BHSP_7:6 for X st the add of X is commutative associative & the add of X has_a_unity & X is_Hilbert_space for S be OrthonormalFamily of X for H be Functional of X st S c= dom H & (for x st x in S holds H.x = (x.|.x)) holds S is summable_set iff S is_summable_set_by H; theorem :: BHSP_7:7 for S be Subset of X st S is non empty & S is summable_set holds (for e be Real st 0 < e ex Y0 be finite Subset of X st Y0 is non empty & Y0 c= S & for Y1 be finite Subset of X st Y0 c= Y1 & Y1 c= S holds abs(((sum(S)).|.(sum(S))) - ((setsum(Y1)).|.(setsum(Y1)))) < e); theorem :: BHSP_7:8 for X st the add of X is commutative associative & the add of X has_a_unity & X is_Hilbert_space for S be OrthonormalFamily of X st S is non empty for H be Functional of X st S c= dom H & (for x st x in S holds H.x = (x.|.x)) holds S is summable_set implies (sum(S)).|.(sum(S)) = sum_byfunc(S, H);

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