Journal of Universal Computer Science, vol. 3, no. 11 (1997), submitted: 8/8/97, accepted: 16/10/97, appeared: 28/11/97 Springer Pub. Co.

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1 Journal of Universal Computer Science, vol. 3, no. 11 (1997), submitted: 8/8/97, accepted: 16/10/97, appeared: 28/11/97 Springer Pub. Co. Sequential Continuity of Linear Mappings in Constructive Mathematics 1 Hajime Ishihara èjapan Advanced Institute of Science and Technology, Japan ishihara@jaist.ac.jpè Abstract: This paper deals, constructively, with two theorems on the sequential continuity of linear mappings. The classical proofs of these theorems use the boundedness of the linear mappings, which is a constructively stronger property than sequential continuity; and constructively inadmissable versions of the Banach-Steinhaus theorem. Key Words: sequential continuity, pointwise continuity, linear mappings, constructive mathematics Category: F.m 1 Introduction The classical validity of many important theorems of analysis, such as the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, depends on Baire's theorem about complete metric spaces, which is an indispensable tool in this area. A form of Baire's theorem has a constructive proof ëbridges and Richman 87, Theorem 1.3ë, but its classical equivalent, If a complete metric space is the union of a sequence of its subsets, then the closure of at least one set in the sequence must have nonempty interior, which is used in the standard argument to prove the above theorems has no known constructive proof. In ëishihara 91, Ishihara 92, Ishihara 93ë, we dealt with the constructive distinctions between certain types of continuity, such as sequential and pointwise continuity; we subsequently proved, in ëishihara 94ë, constructive versions of Banach's inverse mapping theorem, the open mapping theorem, and the closed graph theorem for sequentially continuous linear mappings. In this paper we prove, within Bishop's constructive mathematics, two theorems on the sequential continuity of linear mappings which are classically proved using the boundedness èpointwise continuityè of linear mappings, the Banach-Steinhaus theorem, and the closed graph theorem ërudin 91, Theorem 2.8 and Theorem 5.1ë. We assume that the reader has access to ëbishop and Bridges 85ë or ëbridges and Richman 87ë, but very little background in constructive analysis is required for the understanding of our proofs. 1 Proceedings of the First Japan-New Zealand Workshop on Logic in Computer Science, special issue editors D.S. Bridges, C.S. Calude, M.J. Dinneen and B. Khoussainov.

2 Ishihara H.: Sequential Continuity of Linear Mappings in Constructive Mathematics Main results We begin by proving two lemmas which, despite having been referenced on occasion, have only appeared in a preprint ëishihara 90ë. Lemma 1. Let T be a linear mapping of a Banach space X into a normed space Y, and let èx n è be asequence converging to 0 in X. Then for all positive numbers a; b with aéb, either ktx n k éafor some n or else ktx n k ébfor all n. Proof. Let èn k è 1 k=1 that and, taking n 0 := 0, set be a strictly increasing sequence of positive integers such kx j k é 1 èk +1è 2 èjçn k è; s k := maxfktx j k: n k,1 éjçn k g: Construct an increasing binary sequence èç n è such that ç k =0 è s k éb; ç k =1 è aés k : Wemay assume that ç 1 = 0. Deæne a sequence èy k èinxas follows: if ç k = 0, set y k := 0; if ç k =1,ç k,1,choose j with n k,1 éjçn k and ktx j k éa, and set y i := kx j for all i ç k. Then èy k è is a Cauchy sequence: in fact, ky i, y k kç1=k whenever i ç k. Soèy k è converges to a limit y in X. Choosing a positive integer N such that ktyk éna, consider any integer k ç N. Ifç k =1,ç k,1, then y = kx j for some j with ktx j k éa,so Na ç ka é kktx j k = ktykéna; a contradiction. Hence ç k = ç k,1 for all k ç N. It follows that either ç k = 1 for some k or else ç k = 0 for all k. ut Lemma 2. Let T be a linear mapping of a Banach space X into a normed space Y, and let èx n è be asequence converging to 0 in X. Then for all positive numbers a; b with aéb,either ktx n k éafor inænitely many n or else ktx n k ébfor all suæciently large n. Proof. Let èn k è 1 k=1 that be a strictly increasing sequence of positive integers such kx j k é 1 k 2 èj ç n k è: By successively applying Lemma 1 to the sequences èx nk ;x nk +1;:::è, construct an increasing binary sequence èç k è such that ç k =0 è 9jçn k èaéktx j kè; ç k =1 è 8jçn k èktx j k ébè: Wemay assume that ç 1 = 0. Deæne a sequence èz k èinxas follows: if ç k =0, choose j ç n k such that ktx j k éaand set z k := kx j ; if ç k =1,ç k,1,set z i := kx k,1 for all i ç k. Then èz k è is a Cauchy sequence, and so converges to a

3 1252 Ishihara H.: Sequential Continuity of Linear Mappings in Constructive Mathematics limit z 2 X. Choosing a positive integer N such that ktzkéna, consider any integer kén.ifç k =1,ç k,1, then z =èk,1èx j for some j with ktx j k éa, so Na ç èk,1èa ékèk,1ètx j k= ktykéna; a contradiction. Hence ç k = ç k,1 for all k ç N. It follows that either ç k =1 for some k or else ç k = 0 for all k. In the former case, for each k there exists j ç n k such that ktx j k éa; in the latter, there exists k such that ktx j k éb for all j ç n k. ut A mapping f : X!Y between metric spaces is said to be 1. discontinuous if there exist a sequence èx n è in X, apointx2x, and a positive number æ such that x n!x and dèfèx n è;fèxèè ç æ for all n; and 2. strongly extensional if fèxè 6= fèyè implies x 6= y èwhere x 6= y means 0 édèx; yèè. The following lemma is proved in ëishihara 94ë. Lemma 3. If there exists a mapping f of a complete metric space into a metric space such that f is strongly extensional and discontinuous, then 9-PEM: holds. ut 8æ 2 IN IN è9nèæènè 6= 0è_8nèæènè = 0èè Many well-known theorems in classical analysis could be proved with 9-PEM. For example, one can see that the following lemmas are immediate consequences of 9-PEM and the constructive least-upper-bound principle ëbishop and Bridges 85, è4.4.3èë. A linear mapping T : X!Y between normed spaces is said to be 1. sequentially continuous if x n!0 implies that Tx n!0; 2. bounded if there exists cé0 such that ktxkçckxkfor all x 2 X. Lemma 4. Under 9-PEM, every sequentially continuous linear mapping from a separable normed space into a normed space is bounded. ut A bounded linear mapping T : X!Y between normed spaces is said to be normable if its operator norm exists. kt k := supfkt èxèk : x 2 X; kxk ç1g Lemma 5. Under 9-PEM, every bounded linear functional on a separable normed space is normable. ut Recall that the following form of the uniform boundedness theorem has a constructive proof ëbishop and Bridges 85, page 392, Problem 20ë. Theorem 6. Let èt n è be asequence ofbounded linear mappings from a Banach space X into a normed space Y, and let èx n è be asequence of unit vectors in X such that kt n x n k!1. Then there exists x 2 X such that èkt n xkè is unbounded. ut

4 Ishihara H.: Sequential Continuity of Linear Mappings in Constructive Mathematics 1253 We now prove a constructive version of ërudin 91, Theorem 2.8ë. Theorem 7. Let èt m è be asequence ofsequentially continuous linear mappings from a separable Banach space X into a normed space Y such that Tx := lim m!1 T mx exists for all x 2 X. Then T is sequentially continuous. Proof. Since X is complete, T is strongly extensional, by ëbridges and Ishihara 90, Corollary 2ë. Let èx n è be a sequence converging to 0 in X. Then by Lemma 2, for each æé0 either ktx n k éæ=2 for inænitely many n or else ktx n k éæfor all suæciently large n. In the former case, by passing to an appropriate subsequence, we may assume that ktx n k éæ=2 for all n. Therefore 9-PEM holds, by Lemma 3; so, by Lemma 4, every sequentially continuous linear mapping of X into Y is bounded. Construct a sequence èy n èinxand a strictly increasing sequence èm n è of positive integers such that ky n k = 1 and kt mn y n k!1. Then applying the uniform boundedness theorem to the sequence èt mn è 1 n=1 of bounded linear mappings on X,we can now produce z 2 X such that the sequence èkt mn zkè 1 n=1 is unbounded. This contradicts the hypotheses that Tz = lim m!1 T mz = lim n!1 T mn z exists, and so ensures that ktx n k éæfor all suæciently large n. Since æé0is arbitrary, it follows that T is sequentially continuous. ut Next we prove a constructive version of ërudin 91, Theorem 5.1ë. Theorem 8. Let T be a linear mapping from a Banach space X into a separable normed space Y with the following property: if f is a normable linear functional f on Y, and èx n è converges to 0 in X, then fètx n è!0. Then T is sequentially continuous. Proof. Let èx n è be a sequence converging to 0 in X. Then, as in the proof of Theorem 7, for each æ é 0 either ktx n k é æ=2 for inænitely many n or else ktx n k é æ for all suæciently large n. In the former case, we can construct a sequence èy n è in X such that y n!0 and 0 é kty n k!1. Therefore 9-PEM holds, by Lemma 3; so every linear functional on Y is normable, and Y æ èthe dual of Y è is a Banach space relative to the usual norm, by Lemma 5. By ëbishop and Bridges 85, è7.4.5èë, for each n there exists f n 2 Y æ such that kf n k = 1 and f n èty n è é kty n k=2. Applying the uniform boundedness theorem to the bounded linear functionals f 7! fèty n èony æ,we can now produce g 2 Y æ such that the sequence ègèty n èè is unbounded. This contradicts the hypotheses of our theorem. ut Acknowledgements The author would like to thank Douglas Bridges for many useful comments and suggestions.

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