:: Banach Space of Absolute Summable Real Sequences :: by Yasumasa Suzuki, Noboru Endou and Yasunari Shidama :: :: Received August 8, 2003 :: Copyright (c) 2003-2012 Association of Mizar Users begin definition func the_set_of_l1RealSequences -> Subset of Linear_Space_of_RealSequences means :Def1: :: RSSPACE3:def 1 for x being set holds ( x in it iff ( x in the_set_of_RealSequences & seq_id x is absolutely_summable ) ); existence ex b1 being Subset of Linear_Space_of_RealSequences st for x being set holds ( x in b1 iff ( x in the_set_of_RealSequences & seq_id x is absolutely_summable ) ) proofend; uniqueness for b1, b2 being Subset of Linear_Space_of_RealSequences st ( for x being set holds ( x in b1 iff ( x in the_set_of_RealSequences & seq_id x is absolutely_summable ) ) ) & ( for x being set holds ( x in b2 iff ( x in the_set_of_RealSequences & seq_id x is absolutely_summable ) ) ) holds b1 = b2 proofend; end; :: deftheorem Def1 defines the_set_of_l1RealSequences RSSPACE3:def_1_:_ for b1 being Subset of Linear_Space_of_RealSequences holds ( b1 = the_set_of_l1RealSequences iff for x being set holds ( x in b1 iff ( x in the_set_of_RealSequences & seq_id x is absolutely_summable ) ) ); theorem Th1: :: RSSPACE3:1 for c being Real for seq being Real_Sequence st seq is convergent holds for rseq being Real_Sequence st ( for i being Element of NAT holds rseq . i = abs ((seq . i) - c) ) holds ( rseq is convergent & lim rseq = abs ((lim seq) - c) ) proofend; registration cluster the_set_of_l1RealSequences -> non empty ; coherence not the_set_of_l1RealSequences is empty proofend; end; registration cluster the_set_of_l1RealSequences -> linearly-closed ; coherence the_set_of_l1RealSequences is linearly-closed proofend; end; Lm1: RLSStruct(# the_set_of_l1RealSequences,(Zero_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)) #) is Subspace of Linear_Space_of_RealSequences by RSSPACE:11; registration cluster RLSStruct(# the_set_of_l1RealSequences,(Zero_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)) #) -> right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ; coherence ( RLSStruct(# the_set_of_l1RealSequences,(Zero_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)) #) is Abelian & RLSStruct(# the_set_of_l1RealSequences,(Zero_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)) #) is add-associative & RLSStruct(# the_set_of_l1RealSequences,(Zero_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)) #) is right_zeroed & RLSStruct(# the_set_of_l1RealSequences,(Zero_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)) #) is right_complementable & RLSStruct(# the_set_of_l1RealSequences,(Zero_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)) #) is vector-distributive & RLSStruct(# the_set_of_l1RealSequences,(Zero_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)) #) is scalar-distributive & RLSStruct(# the_set_of_l1RealSequences,(Zero_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)) #) is scalar-associative & RLSStruct(# the_set_of_l1RealSequences,(Zero_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)) #) is scalar-unital ) by RSSPACE:11; end; Lm2: RLSStruct(# the_set_of_l1RealSequences,(Zero_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)) #) is RealLinearSpace ; definition func l_norm -> Function of the_set_of_l1RealSequences,REAL means :Def2: :: RSSPACE3:def 2 for x being set st x in the_set_of_l1RealSequences holds it . x = Sum (abs (seq_id x)); existence ex b1 being Function of the_set_of_l1RealSequences,REAL st for x being set st x in the_set_of_l1RealSequences holds b1 . x = Sum (abs (seq_id x)) proofend; uniqueness for b1, b2 being Function of the_set_of_l1RealSequences,REAL st ( for x being set st x in the_set_of_l1RealSequences holds b1 . x = Sum (abs (seq_id x)) ) & ( for x being set st x in the_set_of_l1RealSequences holds b2 . x = Sum (abs (seq_id x)) ) holds b1 = b2 proofend; end; :: deftheorem Def2 defines l_norm RSSPACE3:def_2_:_ for b1 being Function of the_set_of_l1RealSequences,REAL holds ( b1 = l_norm iff for x being set st x in the_set_of_l1RealSequences holds b1 . x = Sum (abs (seq_id x)) ); registration let X be non empty set ; let Z be Element of X; let A be BinOp of X; let M be Function of [:REAL,X:],X; let N be Function of X,REAL; cluster NORMSTR(# X,Z,A,M,N #) -> non empty ; coherence not NORMSTR(# X,Z,A,M,N #) is empty ; end; theorem :: RSSPACE3:2 for l being NORMSTR st RLSStruct(# the carrier of l, the ZeroF of l, the addF of l, the Mult of l #) is RealLinearSpace holds l is RealLinearSpace by RSSPACE:15; theorem Th3: :: RSSPACE3:3 for rseq being Real_Sequence st ( for n being Element of NAT holds rseq . n = 0 ) holds ( rseq is absolutely_summable & Sum (abs rseq) = 0 ) proofend; theorem Th4: :: RSSPACE3:4 for rseq being Real_Sequence st rseq is absolutely_summable & Sum (abs rseq) = 0 holds for n being Element of NAT holds rseq . n = 0 proofend; theorem Th5: :: RSSPACE3:5 NORMSTR(# the_set_of_l1RealSequences,(Zero_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),l_norm #) is RealLinearSpace by Lm2, RSSPACE:15; definition func l1_Space -> non empty NORMSTR equals :: RSSPACE3:def 3 NORMSTR(# the_set_of_l1RealSequences,(Zero_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),l_norm #); coherence NORMSTR(# the_set_of_l1RealSequences,(Zero_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),l_norm #) is non empty NORMSTR ; end; :: deftheorem defines l1_Space RSSPACE3:def_3_:_ l1_Space = NORMSTR(# the_set_of_l1RealSequences,(Zero_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_l1RealSequences,Linear_Space_of_RealSequences)),l_norm #); begin theorem Th6: :: RSSPACE3:6 ( the carrier of l1_Space = the_set_of_l1RealSequences & ( for x being set holds ( x is VECTOR of l1_Space iff ( x is Real_Sequence & seq_id x is absolutely_summable ) ) ) & 0. l1_Space = Zeroseq & ( for u being VECTOR of l1_Space holds u = seq_id u ) & ( for u, v being VECTOR of l1_Space holds u + v = (seq_id u) + (seq_id v) ) & ( for r being Real for u being VECTOR of l1_Space holds r * u = r (#) (seq_id u) ) & ( for u being VECTOR of l1_Space holds ( - u = - (seq_id u) & seq_id (- u) = - (seq_id u) ) ) & ( for u, v being VECTOR of l1_Space holds u - v = (seq_id u) - (seq_id v) ) & ( for v being VECTOR of l1_Space holds seq_id v is absolutely_summable ) & ( for v being VECTOR of l1_Space holds ||.v.|| = Sum (abs (seq_id v)) ) ) proofend; theorem Th7: :: RSSPACE3:7 for x, y being Point of l1_Space for a being Real holds ( ( ||.x.|| = 0 implies x = 0. l1_Space ) & ( x = 0. l1_Space implies ||.x.|| = 0 ) & 0 <= ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(a * x).|| = (abs a) * ||.x.|| ) proofend; registration cluster l1_Space -> non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ; coherence ( l1_Space is reflexive & l1_Space is discerning & l1_Space is RealNormSpace-like & l1_Space is vector-distributive & l1_Space is scalar-distributive & l1_Space is scalar-associative & l1_Space is scalar-unital & l1_Space is Abelian & l1_Space is add-associative & l1_Space is right_zeroed & l1_Space is right_complementable ) proofend; end; Lm3: for c being Real for seq, seq1 being Real_Sequence st seq is convergent & seq1 is convergent holds for rseq being Real_Sequence st ( for i being Element of NAT holds rseq . i = (abs ((seq . i) - c)) + (seq1 . i) ) holds ( rseq is convergent & lim rseq = (abs ((lim seq) - c)) + (lim seq1) ) proofend; definition let X be non empty NORMSTR ; let x, y be Point of X; func dist (x,y) -> Real equals :: RSSPACE3:def 4 ||.(x - y).||; coherence ||.(x - y).|| is Real ; end; :: deftheorem defines dist RSSPACE3:def_4_:_ for X being non empty NORMSTR for x, y being Point of X holds dist (x,y) = ||.(x - y).||; definition let NRM be non empty NORMSTR ; let seqt be sequence of NRM; attrseqt is CCauchy means :Def5: :: RSSPACE3:def 5 for r1 being Real st r1 > 0 holds ex k1 being Element of NAT st for n1, m1 being Element of NAT st n1 >= k1 & m1 >= k1 holds dist ((seqt . n1),(seqt . m1)) < r1; end; :: deftheorem Def5 defines CCauchy RSSPACE3:def_5_:_ for NRM being non empty NORMSTR for seqt being sequence of NRM holds ( seqt is CCauchy iff for r1 being Real st r1 > 0 holds ex k1 being Element of NAT st for n1, m1 being Element of NAT st n1 >= k1 & m1 >= k1 holds dist ((seqt . n1),(seqt . m1)) < r1 ); notation let NRM be non empty NORMSTR ; let seqt be sequence of NRM; synonym Cauchy_sequence_by_Norm seqt for CCauchy ; end; theorem Th8: :: RSSPACE3:8 for NRM being RealNormSpace for seq being sequence of NRM holds ( seq is Cauchy_sequence_by_Norm iff for r being Real st r > 0 holds ex k being Element of NAT st for n, m being Element of NAT st n >= k & m >= k holds ||.((seq . n) - (seq . m)).|| < r ) proofend; theorem :: RSSPACE3:9 for vseq being sequence of l1_Space st vseq is Cauchy_sequence_by_Norm holds vseq is convergent proofend;