Volume 2, 1990

University of Bialystok

Copyright (c) 1990 Association of Mizar Users

### The abstract of the Mizar article:

### Real Normed Space

**by****Jan Popiolek**- Received September 20, 1990
- MML identifier: NORMSP_1

- [ Mizar article, MML identifier index ]

environ vocabulary RLVECT_1, BINOP_1, FUNCT_1, PRE_TOPC, RLSUB_1, ABSVALUE, ARYTM_1, ARYTM_3, RELAT_1, SEQM_3, SEQ_2, SEQ_1, ORDINAL2, NORMSP_1, ARYTM; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, NUMBERS, XREAL_0, REAL_1, NAT_1, FUNCT_1, FUNCT_2, BINOP_1, RELAT_1, SEQ_1, SEQ_2, SEQM_3, ABSVALUE, STRUCT_0, PRE_TOPC, RLVECT_1, RLSUB_1; constructors REAL_1, NAT_1, DOMAIN_1, SEQ_2, SEQM_3, ABSVALUE, RLSUB_1, MEMBERED, XBOOLE_0; clusters STRUCT_0, XREAL_0, RELSET_1, ARYTM_3, MEMBERED, ZFMISC_1, XBOOLE_0, NUMBERS, ORDINAL2; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; begin definition struct(RLSStruct) NORMSTR (# carrier -> set, Zero -> Element of the carrier, add -> BinOp of the carrier, Mult -> Function of [:REAL, the carrier:], the carrier, norm -> Function of the carrier, REAL #); end; definition cluster non empty NORMSTR; end; reserve X for non empty NORMSTR; reserve a, b for Real; reserve x for Point of X; definition let X, x; func ||.x.|| -> Real equals :: NORMSP_1:def 1 ( the norm of X ).x; end; definition let IT be non empty NORMSTR; attr IT is RealNormSpace-like means :: NORMSP_1:def 2 for x , y being Point of IT , a holds ( ||.x.|| = 0 iff x = 0.IT ) & ||.a * x.|| = abs(a) * ||.x.|| & ||.x + y.|| <= ||.x.|| + ||.y.||; end; definition cluster RealNormSpace-like RealLinearSpace-like Abelian add-associative right_zeroed right_complementable strict (non empty NORMSTR); end; definition mode RealNormSpace is RealNormSpace-like RealLinearSpace-like Abelian add-associative right_zeroed right_complementable (non empty NORMSTR); end; reserve RNS for RealNormSpace; reserve x, y, z, g, g1, g2 for Point of RNS; canceled 4; theorem :: NORMSP_1:5 ||.0.RNS.|| = 0; theorem :: NORMSP_1:6 ||.-x.|| = ||.x.||; theorem :: NORMSP_1:7 ||.x - y.|| <= ||.x.|| + ||.y.||; theorem :: NORMSP_1:8 0 <= ||.x.||; theorem :: NORMSP_1:9 ||.a * x + b * y.|| <= abs(a) * ||.x.|| + abs(b) * ||.y.||; theorem :: NORMSP_1:10 ||.x - y.|| = 0 iff x = y; theorem :: NORMSP_1:11 ||.x - y.|| = ||.y - x.||; theorem :: NORMSP_1:12 ||.x.|| - ||.y.|| <= ||.x - y.||; theorem :: NORMSP_1:13 abs(||.x.|| - ||.y.||) <= ||.x - y.||; theorem :: NORMSP_1:14 ||.x - z.|| <= ||.x - y.|| + ||.y - z.||; theorem :: NORMSP_1:15 x <> y implies ||.x - y.|| <> 0; definition let RNS be 1-sorted; mode sequence of RNS means :: NORMSP_1:def 3 it is Function of NAT, the carrier of RNS; end; definition let RNS be 1-sorted; cluster -> Function-like Relation-like sequence of RNS; end; definition let RNS be non empty 1-sorted; redefine mode sequence of RNS -> Function of NAT, the carrier of RNS; end; reserve S, S1, S2 for sequence of RNS; reserve k, n, m, m1, m2 for Nat; reserve r for Real; reserve f for Function; reserve d, s, t for set; canceled; theorem :: NORMSP_1:17 for RNS being non empty 1-sorted, x being Element of RNS holds f is sequence of RNS iff ( dom f = NAT & for d st d in NAT holds f.d is Element of RNS ); canceled; theorem :: NORMSP_1:19 for RNS being non empty 1-sorted, x being Element of RNS ex S being sequence of RNS st rng S = {x}; theorem :: NORMSP_1:20 for RNS being non empty 1-sorted, S being sequence of RNS st (ex x being Element of RNS st for n holds S.n = x) ex x being Element of RNS st rng S={x}; theorem :: NORMSP_1:21 for RNS being non empty 1-sorted, S being sequence of RNS st ex x being Element of RNS st rng S = {x} holds for n holds S.n = S.(n+1); theorem :: NORMSP_1:22 for RNS being non empty 1-sorted, S being sequence of RNS st for n holds S.n = S.(n+1) holds for n , k holds S.n = S.(n+k); theorem :: NORMSP_1:23 for RNS being non empty 1-sorted, S being sequence of RNS st for n , k holds S.n = S.(n+k) holds for n , m holds S.n = S.m; theorem :: NORMSP_1:24 for RNS being non empty 1-sorted, S being sequence of RNS st for n , m holds S.n = S.m ex x being Element of RNS st for n holds S.n = x; theorem :: NORMSP_1:25 ex S st rng S = {0.RNS}; definition let RNS be non empty 1-sorted; let S be sequence of RNS; redefine attr S is constant means :: NORMSP_1:def 4 ex x being Element of RNS st for n holds S.n = x; end; canceled; theorem :: NORMSP_1:27 for RNS being non empty 1-sorted, S being sequence of RNS holds S is constant iff ex x being Element of RNS st rng S = {x}; definition let RNS be non empty 1-sorted; let S be sequence of RNS; let n; redefine func S.n -> Element of RNS; end; scheme ExRNSSeq{RNS()->RealNormSpace, F(Nat) -> ( Point of RNS() ) } : ex S be sequence of RNS() st for n holds S.n = F(n); scheme ExRLSSeq{RNS()->RealLinearSpace, F(Nat) -> Element of RNS() } : ex S be sequence of RNS() st for n holds S.n = F(n); definition let RNS be RealLinearSpace; let S1, S2 be sequence of RNS; func S1 + S2 -> sequence of RNS means :: NORMSP_1:def 5 for n holds it.n = S1.n + S2.n; end; definition let RNS be RealLinearSpace; let S1 , S2 be sequence of RNS; func S1 - S2 -> sequence of RNS means :: NORMSP_1:def 6 for n holds it.n = S1.n - S2.n; end; definition let RNS be RealLinearSpace; let S be sequence of RNS; let x be Element of RNS; func S - x -> sequence of RNS means :: NORMSP_1:def 7 for n holds it.n = S.n - x; end; definition let RNS be RealLinearSpace; let S be sequence of RNS; let a; func a * S -> sequence of RNS means :: NORMSP_1:def 8 for n holds it.n = a * S.n; end; definition let RNS; let S; attr S is convergent means :: NORMSP_1:def 9 ex g st for r st 0 < r ex m st for n st m <= n holds ||.(S.n) - g.|| < r; end; canceled 6; theorem :: NORMSP_1:34 S1 is convergent & S2 is convergent implies S1 + S2 is convergent; theorem :: NORMSP_1:35 S1 is convergent & S2 is convergent implies S1 - S2 is convergent; theorem :: NORMSP_1:36 S is convergent implies S - x is convergent; theorem :: NORMSP_1:37 S is convergent implies a * S is convergent; definition let RNS; let S; func ||.S.|| -> Real_Sequence means :: NORMSP_1:def 10 for n holds it.n =||.(S.n).||; end; canceled; theorem :: NORMSP_1:39 S is convergent implies ||.S.|| is convergent; definition let RNS; let S; assume S is convergent; func lim S -> Point of RNS means :: NORMSP_1:def 11 for r st 0 < r ex m st for n st m <= n holds ||.(S.n) - it.|| < r; end; canceled; theorem :: NORMSP_1:41 S is convergent & lim S = g implies ( ||.S - g.|| is convergent & lim ||.S - g.|| = 0 ); theorem :: NORMSP_1:42 S1 is convergent & S2 is convergent implies lim (S1 + S2) = (lim S1) + (lim S2); theorem :: NORMSP_1:43 S1 is convergent & S2 is convergent implies lim (S1 - S2) = (lim S1) - (lim S2); theorem :: NORMSP_1:44 S is convergent implies lim (S - x) = (lim S) - x; theorem :: NORMSP_1:45 S is convergent implies lim (a * S) = a * (lim S);

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