:: Banach Space of Bounded Real Sequences :: by Yasumasa Suzuki :: :: Received January 6, 2004 :: Copyright (c) 2004-2012 Association of Mizar Users begin Lm1: for rseq being Real_Sequence for K being real number st ( for n being Element of NAT holds rseq . n <= K ) holds upper_bound (rng rseq) <= K proofend; Lm2: for rseq being Real_Sequence st rseq is bounded holds for n being Element of NAT holds rseq . n <= upper_bound (rng rseq) proofend; definition func the_set_of_BoundedRealSequences -> Subset of Linear_Space_of_RealSequences means :Def1: :: RSSPACE4:def 1 for x being set holds ( x in it iff ( x in the_set_of_RealSequences & seq_id x is bounded ) ); 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 bounded ) ) 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 bounded ) ) ) & ( for x being set holds ( x in b2 iff ( x in the_set_of_RealSequences & seq_id x is bounded ) ) ) holds b1 = b2 proofend; end; :: deftheorem Def1 defines the_set_of_BoundedRealSequences RSSPACE4:def_1_:_ for b1 being Subset of Linear_Space_of_RealSequences holds ( b1 = the_set_of_BoundedRealSequences iff for x being set holds ( x in b1 iff ( x in the_set_of_RealSequences & seq_id x is bounded ) ) ); registration cluster the_set_of_BoundedRealSequences -> non empty ; coherence not the_set_of_BoundedRealSequences is empty proofend; cluster the_set_of_BoundedRealSequences -> linearly-closed ; coherence the_set_of_BoundedRealSequences is linearly-closed proofend; end; Lm3: RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is Subspace of Linear_Space_of_RealSequences by RSSPACE:11; registration cluster RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) -> right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ; coherence ( RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is Abelian & RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is add-associative & RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is right_zeroed & RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is right_complementable & RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is vector-distributive & RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is scalar-distributive & RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is scalar-associative & RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is scalar-unital ) by RSSPACE:11; end; Lm4: ( RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is Abelian & RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is add-associative & RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is right_zeroed & RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is right_complementable & RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is vector-distributive & RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is scalar-distributive & RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is scalar-associative & RLSStruct(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)) #) is scalar-unital ) ; definition func linfty_norm -> Function of the_set_of_BoundedRealSequences,REAL means :Def2: :: RSSPACE4:def 2 for x being set st x in the_set_of_BoundedRealSequences holds it . x = upper_bound (rng (abs (seq_id x))); existence ex b1 being Function of the_set_of_BoundedRealSequences,REAL st for x being set st x in the_set_of_BoundedRealSequences holds b1 . x = upper_bound (rng (abs (seq_id x))) proofend; uniqueness for b1, b2 being Function of the_set_of_BoundedRealSequences,REAL st ( for x being set st x in the_set_of_BoundedRealSequences holds b1 . x = upper_bound (rng (abs (seq_id x))) ) & ( for x being set st x in the_set_of_BoundedRealSequences holds b2 . x = upper_bound (rng (abs (seq_id x))) ) holds b1 = b2 proofend; end; :: deftheorem Def2 defines linfty_norm RSSPACE4:def_2_:_ for b1 being Function of the_set_of_BoundedRealSequences,REAL holds ( b1 = linfty_norm iff for x being set st x in the_set_of_BoundedRealSequences holds b1 . x = upper_bound (rng (abs (seq_id x))) ); Lm5: for rseq being Real_Sequence st ( for n being Nat holds rseq . n = 0 ) holds ( rseq is bounded & upper_bound (rng (abs rseq)) = 0 ) proofend; Lm6: for rseq being Real_Sequence st rseq is bounded & upper_bound (rng (abs rseq)) = 0 holds for n being Nat holds rseq . n = 0 proofend; theorem :: RSSPACE4:1 for rseq being Real_Sequence holds ( ( rseq is bounded & upper_bound (rng (abs rseq)) = 0 ) iff for n being Nat holds rseq . n = 0 ) by Lm5, Lm6; registration cluster NORMSTR(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),linfty_norm #) -> right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ; coherence ( NORMSTR(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),linfty_norm #) is Abelian & NORMSTR(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),linfty_norm #) is add-associative & NORMSTR(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),linfty_norm #) is right_zeroed & NORMSTR(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),linfty_norm #) is right_complementable & NORMSTR(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),linfty_norm #) is vector-distributive & NORMSTR(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),linfty_norm #) is scalar-distributive & NORMSTR(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),linfty_norm #) is scalar-associative & NORMSTR(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),linfty_norm #) is scalar-unital ) by Lm4, RSSPACE3:2; end; definition func linfty_Space -> non empty NORMSTR equals :: RSSPACE4:def 3 NORMSTR(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),linfty_norm #); coherence NORMSTR(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),linfty_norm #) is non empty NORMSTR ; end; :: deftheorem defines linfty_Space RSSPACE4:def_3_:_ linfty_Space = NORMSTR(# the_set_of_BoundedRealSequences,(Zero_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Add_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),(Mult_ (the_set_of_BoundedRealSequences,Linear_Space_of_RealSequences)),linfty_norm #); theorem Th2: :: RSSPACE4:2 ( the carrier of linfty_Space = the_set_of_BoundedRealSequences & ( for x being set holds ( x is VECTOR of linfty_Space iff ( x is Real_Sequence & seq_id x is bounded ) ) ) & 0. linfty_Space = Zeroseq & ( for u being VECTOR of linfty_Space holds u = seq_id u ) & ( for u, v being VECTOR of linfty_Space holds u + v = (seq_id u) + (seq_id v) ) & ( for r being Real for u being VECTOR of linfty_Space holds r * u = r (#) (seq_id u) ) & ( for u being VECTOR of linfty_Space holds ( - u = - (seq_id u) & seq_id (- u) = - (seq_id u) ) ) & ( for u, v being VECTOR of linfty_Space holds u - v = (seq_id u) - (seq_id v) ) & ( for v being VECTOR of linfty_Space holds seq_id v is bounded ) & ( for v being VECTOR of linfty_Space holds ||.v.|| = upper_bound (rng (abs (seq_id v))) ) ) proofend; theorem Th3: :: RSSPACE4:3 for x, y being Point of linfty_Space for a being Real holds ( ( ||.x.|| = 0 implies x = 0. linfty_Space ) & ( x = 0. linfty_Space implies ||.x.|| = 0 ) & 0 <= ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(a * x).|| = (abs a) * ||.x.|| ) proofend; registration cluster linfty_Space -> non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ; coherence ( linfty_Space is reflexive & linfty_Space is discerning & linfty_Space is RealNormSpace-like & linfty_Space is vector-distributive & linfty_Space is scalar-distributive & linfty_Space is scalar-associative & linfty_Space is scalar-unital & linfty_Space is Abelian & linfty_Space is add-associative & linfty_Space is right_zeroed & linfty_Space is right_complementable ) proofend; end; theorem :: RSSPACE4:4 for vseq being sequence of linfty_Space st vseq is Cauchy_sequence_by_Norm holds vseq is convergent proofend; begin definition let X be non empty set ; let Y be RealNormSpace; let IT be Function of X, the carrier of Y; attrIT is bounded means :Def4: :: RSSPACE4:def 4 ex K being Real st ( 0 <= K & ( for x being Element of X holds ||.(IT . x).|| <= K ) ); end; :: deftheorem Def4 defines bounded RSSPACE4:def_4_:_ for X being non empty set for Y being RealNormSpace for IT being Function of X, the carrier of Y holds ( IT is bounded iff ex K being Real st ( 0 <= K & ( for x being Element of X holds ||.(IT . x).|| <= K ) ) ); theorem Th5: :: RSSPACE4:5 for X being non empty set for Y being RealNormSpace for f being Function of X, the carrier of Y st ( for x being Element of X holds f . x = 0. Y ) holds f is bounded proofend; registration let X be non empty set ; let Y be RealNormSpace; cluster non empty Relation-like X -defined the carrier of Y -valued Function-like V23(X) quasi_total bounded for Element of bool [:X, the carrier of Y:]; existence ex b1 being Function of X, the carrier of Y st b1 is bounded proofend; end; definition let X be non empty set ; let Y be RealNormSpace; func BoundedFunctions (X,Y) -> Subset of (RealVectSpace (X,Y)) means :Def5: :: RSSPACE4:def 5 for x being set holds ( x in it iff x is bounded Function of X, the carrier of Y ); existence ex b1 being Subset of (RealVectSpace (X,Y)) st for x being set holds ( x in b1 iff x is bounded Function of X, the carrier of Y ) proofend; uniqueness for b1, b2 being Subset of (RealVectSpace (X,Y)) st ( for x being set holds ( x in b1 iff x is bounded Function of X, the carrier of Y ) ) & ( for x being set holds ( x in b2 iff x is bounded Function of X, the carrier of Y ) ) holds b1 = b2 proofend; end; :: deftheorem Def5 defines BoundedFunctions RSSPACE4:def_5_:_ for X being non empty set for Y being RealNormSpace for b3 being Subset of (RealVectSpace (X,Y)) holds ( b3 = BoundedFunctions (X,Y) iff for x being set holds ( x in b3 iff x is bounded Function of X, the carrier of Y ) ); registration let X be non empty set ; let Y be RealNormSpace; cluster BoundedFunctions (X,Y) -> non empty ; coherence not BoundedFunctions (X,Y) is empty proofend; end; theorem Th6: :: RSSPACE4:6 for X being non empty set for Y being RealNormSpace holds BoundedFunctions (X,Y) is linearly-closed proofend; theorem :: RSSPACE4:7 for X being non empty set for Y being RealNormSpace holds RLSStruct(# (BoundedFunctions (X,Y)),(Zero_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Add_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Mult_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))) #) is Subspace of RealVectSpace (X,Y) by Th6, RSSPACE:11; registration let X be non empty set ; let Y be RealNormSpace; cluster RLSStruct(# (BoundedFunctions (X,Y)),(Zero_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Add_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Mult_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))) #) -> right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ; coherence ( RLSStruct(# (BoundedFunctions (X,Y)),(Zero_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Add_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Mult_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))) #) is Abelian & RLSStruct(# (BoundedFunctions (X,Y)),(Zero_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Add_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Mult_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))) #) is add-associative & RLSStruct(# (BoundedFunctions (X,Y)),(Zero_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Add_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Mult_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))) #) is right_zeroed & RLSStruct(# (BoundedFunctions (X,Y)),(Zero_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Add_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Mult_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))) #) is right_complementable & RLSStruct(# (BoundedFunctions (X,Y)),(Zero_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Add_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Mult_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))) #) is vector-distributive & RLSStruct(# (BoundedFunctions (X,Y)),(Zero_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Add_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Mult_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))) #) is scalar-distributive & RLSStruct(# (BoundedFunctions (X,Y)),(Zero_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Add_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Mult_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))) #) is scalar-associative & RLSStruct(# (BoundedFunctions (X,Y)),(Zero_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Add_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Mult_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))) #) is scalar-unital ) by Th6, RSSPACE:11; end; definition let X be non empty set ; let Y be RealNormSpace; func R_VectorSpace_of_BoundedFunctions (X,Y) -> RealLinearSpace equals :: RSSPACE4:def 6 RLSStruct(# (BoundedFunctions (X,Y)),(Zero_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Add_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Mult_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))) #); coherence RLSStruct(# (BoundedFunctions (X,Y)),(Zero_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Add_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Mult_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))) #) is RealLinearSpace ; end; :: deftheorem defines R_VectorSpace_of_BoundedFunctions RSSPACE4:def_6_:_ for X being non empty set for Y being RealNormSpace holds R_VectorSpace_of_BoundedFunctions (X,Y) = RLSStruct(# (BoundedFunctions (X,Y)),(Zero_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Add_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Mult_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))) #); registration let X be non empty set ; let Y be RealNormSpace; cluster R_VectorSpace_of_BoundedFunctions (X,Y) -> strict ; coherence R_VectorSpace_of_BoundedFunctions (X,Y) is strict ; end; theorem Th8: :: RSSPACE4:8 for X being non empty set for Y being RealNormSpace for f, g, h being VECTOR of (R_VectorSpace_of_BoundedFunctions (X,Y)) for f9, g9, h9 being bounded Function of X, the carrier of Y st f9 = f & g9 = g & h9 = h holds ( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) ) proofend; theorem Th9: :: RSSPACE4:9 for X being non empty set for Y being RealNormSpace for f, h being VECTOR of (R_VectorSpace_of_BoundedFunctions (X,Y)) for f9, h9 being bounded Function of X, the carrier of Y st f9 = f & h9 = h holds for a being Real holds ( h = a * f iff for x being Element of X holds h9 . x = a * (f9 . x) ) proofend; theorem Th10: :: RSSPACE4:10 for X being non empty set for Y being RealNormSpace holds 0. (R_VectorSpace_of_BoundedFunctions (X,Y)) = X --> (0. Y) proofend; definition let X be non empty set ; let Y be RealNormSpace; let f be set ; assume A1: f in BoundedFunctions (X,Y) ; func modetrans (f,X,Y) -> bounded Function of X, the carrier of Y equals :Def7: :: RSSPACE4:def 7 f; coherence f is bounded Function of X, the carrier of Y by A1, Def5; end; :: deftheorem Def7 defines modetrans RSSPACE4:def_7_:_ for X being non empty set for Y being RealNormSpace for f being set st f in BoundedFunctions (X,Y) holds modetrans (f,X,Y) = f; definition let X be non empty set ; let Y be RealNormSpace; let u be Function of X, the carrier of Y; func PreNorms u -> non empty Subset of REAL equals :: RSSPACE4:def 8 { ||.(u . t).|| where t is Element of X : verum } ; coherence { ||.(u . t).|| where t is Element of X : verum } is non empty Subset of REAL proofend; end; :: deftheorem defines PreNorms RSSPACE4:def_8_:_ for X being non empty set for Y being RealNormSpace for u being Function of X, the carrier of Y holds PreNorms u = { ||.(u . t).|| where t is Element of X : verum } ; theorem Th11: :: RSSPACE4:11 for X being non empty set for Y being RealNormSpace for g being bounded Function of X, the carrier of Y holds PreNorms g is bounded_above proofend; theorem :: RSSPACE4:12 for X being non empty set for Y being RealNormSpace for g being Function of X, the carrier of Y holds ( g is bounded iff PreNorms g is bounded_above ) proofend; definition let X be non empty set ; let Y be RealNormSpace; func BoundedFunctionsNorm (X,Y) -> Function of (BoundedFunctions (X,Y)),REAL means :Def9: :: RSSPACE4:def 9 for x being set st x in BoundedFunctions (X,Y) holds it . x = upper_bound (PreNorms (modetrans (x,X,Y))); existence ex b1 being Function of (BoundedFunctions (X,Y)),REAL st for x being set st x in BoundedFunctions (X,Y) holds b1 . x = upper_bound (PreNorms (modetrans (x,X,Y))) proofend; uniqueness for b1, b2 being Function of (BoundedFunctions (X,Y)),REAL st ( for x being set st x in BoundedFunctions (X,Y) holds b1 . x = upper_bound (PreNorms (modetrans (x,X,Y))) ) & ( for x being set st x in BoundedFunctions (X,Y) holds b2 . x = upper_bound (PreNorms (modetrans (x,X,Y))) ) holds b1 = b2 proofend; end; :: deftheorem Def9 defines BoundedFunctionsNorm RSSPACE4:def_9_:_ for X being non empty set for Y being RealNormSpace for b3 being Function of (BoundedFunctions (X,Y)),REAL holds ( b3 = BoundedFunctionsNorm (X,Y) iff for x being set st x in BoundedFunctions (X,Y) holds b3 . x = upper_bound (PreNorms (modetrans (x,X,Y))) ); theorem Th13: :: RSSPACE4:13 for X being non empty set for Y being RealNormSpace for f being bounded Function of X, the carrier of Y holds modetrans (f,X,Y) = f proofend; theorem Th14: :: RSSPACE4:14 for X being non empty set for Y being RealNormSpace for f being bounded Function of X, the carrier of Y holds (BoundedFunctionsNorm (X,Y)) . f = upper_bound (PreNorms f) proofend; definition let X be non empty set ; let Y be RealNormSpace; func R_NormSpace_of_BoundedFunctions (X,Y) -> non empty NORMSTR equals :: RSSPACE4:def 10 NORMSTR(# (BoundedFunctions (X,Y)),(Zero_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Add_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Mult_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(BoundedFunctionsNorm (X,Y)) #); coherence NORMSTR(# (BoundedFunctions (X,Y)),(Zero_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Add_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Mult_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(BoundedFunctionsNorm (X,Y)) #) is non empty NORMSTR ; end; :: deftheorem defines R_NormSpace_of_BoundedFunctions RSSPACE4:def_10_:_ for X being non empty set for Y being RealNormSpace holds R_NormSpace_of_BoundedFunctions (X,Y) = NORMSTR(# (BoundedFunctions (X,Y)),(Zero_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Add_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(Mult_ ((BoundedFunctions (X,Y)),(RealVectSpace (X,Y)))),(BoundedFunctionsNorm (X,Y)) #); theorem Th15: :: RSSPACE4:15 for X being non empty set for Y being RealNormSpace holds X --> (0. Y) = 0. (R_NormSpace_of_BoundedFunctions (X,Y)) proofend; theorem Th16: :: RSSPACE4:16 for X being non empty set for Y being RealNormSpace for f being Point of (R_NormSpace_of_BoundedFunctions (X,Y)) for g being bounded Function of X, the carrier of Y st g = f holds for t being Element of X holds ||.(g . t).|| <= ||.f.|| proofend; theorem :: RSSPACE4:17 for X being non empty set for Y being RealNormSpace for f being Point of (R_NormSpace_of_BoundedFunctions (X,Y)) holds 0 <= ||.f.|| proofend; theorem Th18: :: RSSPACE4:18 for X being non empty set for Y being RealNormSpace for f being Point of (R_NormSpace_of_BoundedFunctions (X,Y)) st f = 0. (R_NormSpace_of_BoundedFunctions (X,Y)) holds 0 = ||.f.|| proofend; theorem Th19: :: RSSPACE4:19 for X being non empty set for Y being RealNormSpace for f, g, h being Point of (R_NormSpace_of_BoundedFunctions (X,Y)) for f9, g9, h9 being bounded Function of X, the carrier of Y st f9 = f & g9 = g & h9 = h holds ( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) ) proofend; theorem Th20: :: RSSPACE4:20 for X being non empty set for Y being RealNormSpace for f, h being Point of (R_NormSpace_of_BoundedFunctions (X,Y)) for f9, h9 being bounded Function of X, the carrier of Y st f9 = f & h9 = h holds for a being Real holds ( h = a * f iff for x being Element of X holds h9 . x = a * (f9 . x) ) proofend; theorem Th21: :: RSSPACE4:21 for X being non empty set for Y being RealNormSpace for f, g being Point of (R_NormSpace_of_BoundedFunctions (X,Y)) for a being Real holds ( ( ||.f.|| = 0 implies f = 0. (R_NormSpace_of_BoundedFunctions (X,Y)) ) & ( f = 0. (R_NormSpace_of_BoundedFunctions (X,Y)) implies ||.f.|| = 0 ) & ||.(a * f).|| = (abs a) * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| ) proofend; theorem Th22: :: RSSPACE4:22 for X being non empty set for Y being RealNormSpace holds ( R_NormSpace_of_BoundedFunctions (X,Y) is reflexive & R_NormSpace_of_BoundedFunctions (X,Y) is discerning & R_NormSpace_of_BoundedFunctions (X,Y) is RealNormSpace-like ) proofend; theorem Th23: :: RSSPACE4:23 for X being non empty set for Y being RealNormSpace holds R_NormSpace_of_BoundedFunctions (X,Y) is RealNormSpace proofend; registration let X be non empty set ; let Y be RealNormSpace; cluster R_NormSpace_of_BoundedFunctions (X,Y) -> non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ; coherence ( R_NormSpace_of_BoundedFunctions (X,Y) is reflexive & R_NormSpace_of_BoundedFunctions (X,Y) is discerning & R_NormSpace_of_BoundedFunctions (X,Y) is RealNormSpace-like & R_NormSpace_of_BoundedFunctions (X,Y) is vector-distributive & R_NormSpace_of_BoundedFunctions (X,Y) is scalar-distributive & R_NormSpace_of_BoundedFunctions (X,Y) is scalar-associative & R_NormSpace_of_BoundedFunctions (X,Y) is scalar-unital & R_NormSpace_of_BoundedFunctions (X,Y) is Abelian & R_NormSpace_of_BoundedFunctions (X,Y) is add-associative & R_NormSpace_of_BoundedFunctions (X,Y) is right_zeroed & R_NormSpace_of_BoundedFunctions (X,Y) is right_complementable ) by Th23; end; theorem Th24: :: RSSPACE4:24 for X being non empty set for Y being RealNormSpace for f, g, h being Point of (R_NormSpace_of_BoundedFunctions (X,Y)) for f9, g9, h9 being bounded Function of X, the carrier of Y st f9 = f & g9 = g & h9 = h holds ( h = f - g iff for x being Element of X holds h9 . x = (f9 . x) - (g9 . x) ) proofend; Lm7: for e being Real for seq being Real_Sequence st seq is convergent & ex k being Element of NAT st for i being Element of NAT st k <= i holds seq . i <= e holds lim seq <= e proofend; theorem Th25: :: RSSPACE4:25 for X being non empty set for Y being RealNormSpace st Y is complete holds for seq being sequence of (R_NormSpace_of_BoundedFunctions (X,Y)) st seq is Cauchy_sequence_by_Norm holds seq is convergent proofend; theorem Th26: :: RSSPACE4:26 for X being non empty set for Y being RealBanachSpace holds R_NormSpace_of_BoundedFunctions (X,Y) is RealBanachSpace proofend; registration let X be non empty set ; let Y be RealBanachSpace; cluster R_NormSpace_of_BoundedFunctions (X,Y) -> non empty complete ; coherence R_NormSpace_of_BoundedFunctions (X,Y) is complete by Th26; end;