Kučera, Jan

McKennon, Kelly

English

Institute of Mathematics AS CR, v. v. i.

Czechoslovak Mathematical JournalJan Kučera; Kelly McKennonPreservation of fast completeness under inductive limitsCzechoslovak Mathematical Journal, Vol. 42 (1992), No. 1, 15–18Persistent URL: http://dml.cz/dmlcz/128301Terms of use:© Institute of Mathematics AS CR, 1992Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documentsstrictly for personal use. Each copy of any part of this document must contain these Terms of use.This document has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://dml.czCzechoslovak Mathematical Journal, 42 (117) 1992, Praha PRESERVATION O F FAST C O M P L E T E N E S S UNDER INDUCTIVE LIMITS JAN KUSERA, KELLY MCKENNON, Pullman (Received September 13, 1989) A topological vector space V is said to be fast complete if each bounded set is bounded in some continuously embedded Banach space. Here we are concerned with the problem of when an inductive limit of fast complete locally convex spaces is fast complete. Let E be a Hausdorff locally convex inductive limit of a nested sequence of Haus-dorff locally convex spaces En. Then E is said to be regular provided tha t each bounded subset of E is contained and bounded in one of the constituent spaces En. For instance it is known tha t an inductive limit of Hilbert spaces must always be regular [1], while there are inductive limits of reflexive Frechet spaces which are not [2]. Our present results are concerned with inductive limits which satisfy the following property BP: each space En has at least one open subset which is bounded in some subse-quent space Fn+fc-We note tha t in particular inductive limits of Banach spaces satisfy the property BP. Before commencing we recall tha t a set S is said to be nowhere dense in another set T provided the closure of S D T in T has void interior in T—otherwise S is somewhere dense in T. L e m m a . Let E satisfy the property BP. Let N be a Banach space continuously embedded in E. Then there exists at least one m E N such that each non-void open subset of Em is somewhere dense in N. P r o o f . Assume false. In view of the property BP, the space E\ possesses a balanced, convex, neighborhood V\ of the origin which is bounded in some later space K5(i)—by making V\ smaller if necessary, we may assume that it is nowhere dense in iV. 15 Suppose tha t an increasing sequence {S(j)}^=1 of natural numbers has been chosen along with a sequence {Vj}J= 1 of subsets such tha t each Vj is a balanced, convex, neighborhood of the origin in Ks(j-i) (where 5(0) = 1), Vj is nowhere dense in N, and Vj is bounded in Es(j)- We then apply the property BP to select a balanced, convex, neighborhood Vn+\ of the origin on Es(n) and a natural number S(n + 1 ) greated than S(n) such tha t Vn+i is bounded in Ks(n+i), being careful to make our selection so tha t Vn+\ is nowhere dense in N. Having chosen our sequence { V n } ^ ! inductively, we let W be the convex hull of oo the set theoretic union (J Vn. Evidently W is a neighborhood of the origin in E. n = l For each m £ N we construct a point pm in N as follows. Since V\ is nowhere dense in N, there exists a ball B(x\) of radius less than | contained in the unit ball of N such tha t B(x\) does not intersect the closure of mV\ in N. Suppose tha t a sequence {XJ}^=1 has been chosen along with a sequence {B(XJ)}^=1 of balls such tha t , for each j = 1 , . . . , n, B(XJ) does not intersect m(Vi + . . . + Vj), the radius of B(XJ) is less than 2~ J , and (if j > 1) the closure of B(XJ) is contained in B(XJ_\). Note tha t V\ is bounded in Fs(i) a n d so Vi + V2 is contained in a scalar multiple of V2. Similarly V\ + V2 is bounded in V3 and so V\ + V2 + V3 is contained a scalar multiple of V3. Evidently V\ + . . . + Vn is contained in a scalar multiple of Vn, and so m(Vi + . . . + Vn) is as well. Since Vn is nowhere dense in N, it follows tha t m(V\ + . . . + Vn) is as well. Hence there exists an open ball B(xn+\) contained in B(xn) which does not intersect m(Vi + . . . + Vn). By reducing the radius if necessary, we may select B(xn+\) such that its closure is in B(xn) and its radius is less than 2 - ( n + l) The sequence xn is Cauchy and so has a limit pm which lies in the intersection of all the balls B(xn). This point pm does not lie in any of the sets m(Vi + . . . + Vn). Consequently pm is not in the neighborhood mW. Since pm is in the unit ball of N for each m £ N, it follows tha t W does not absorb the unit ball of N. By hypothesis N is continuously embedded and so its unit ball is bounded in E: an absurdity. • T h e o r e m . Let E be a Hausdorff inductive limit of fast complete spaces satisfying property BP. Then the following are equivalent: (\) E is fast complete; (ii) E is a regular inductive limit. P r o o f . We first suppose that (ii) holds and show tha t (i) follows. Let D be any bounded subset of E. Regularity implies that D is a bounded subset of En for some n £ N. Since En is fast complete, D is a bounded subset of some Banach space N continuously embedded in En. Since En is continuously embedded in K, N is as well. Thus E is fast complete. Now we show tha t (i) implies (ii) by assuming that (i) holds and (ii) does not hold, and deriving absurdity. Since (ii) does not hold there exists a bounded subset 16 D of E which is not bounded in any of the constituent spaces En. Since (i) holds, we may assume tha t D is in the closed unit ball of some Banach space N which is continuously embedded in E. From the preceding lemma follows tha t there exists some m G N such tha t each open subset of Em is somewhere dense in N. Because of the property BP we may select a balanced, convex neighborhood V of the origin in Em and an integer k > m such tha t V is bounded in K*. Since V is somewhere dense in N, the closure V of V in N has non-void interior V° in N. Since V is balanced, V° is also balanced and so must be a neighborhood of the origin in N. Hence there exists r > 0 such tha t D C r V ° . Let p G D be arbitrary. Then there exists x\ € rV such tha t l l * i - P | | < 2 ' Suppose tha t the sequence {XJ}"=1 has been chosen such tha t з XJ Є 2~j+1rV and 52 XІ 1 » = i < 2 " ' for each j = 1 , . . . , n. Then [p - 52 xj) € 2~nD C 2 " n r V ° and so there exists Zn+i G 2 - f W such tha t Zn + l (p-t ч) 4 j=l ' (P-Ł*І) < 2 - ( n + l ) _ Evidently we have p = 52 xn m the Banach space N. Let | | denote the n = l Minkowski functional on Em genereted by the neighborhood rV. For each n G N n we have | x n | ^ 2- n + 1 , whence follows t h a t the sequence 52 xj °f Pa-*tial sums is i = i n | | -Cauchy. Since rV is bounded subset of K*, it follows t h a t the sequence 52 xj l s i=i n Cauchy in Kjt. Since Ek is fast-complete and the Cauchy sequence 52 x i 1S bounded, i = i there exists a Banach space M continuously embedded in Ek such t h a t rV is in the n unit ball of M. Since | | dominates the norm on M , it follows t h a t 52 xj *s a bso-i = i n lutely convergent in M and has a limit in the unit ball of M. Thus 52 xj converges i = i n to q in Ek and so to q in E as well. But 52 xj converges to p in N (and thus in E). i = i Since E is Hausdorff, we have p = q. Since p was arbitrary, it follows t h a t D is a subset of the unit ball of M, and thus D is bounded in Ek. This implies t h a t E is regular. • References [1] J. Kucera, K. McKennon: Dieudonne-Schwartz Theorem on Bounded Sets in Inductive Limits, Proc. A.M.S. 78(1980), 366-368. 17 [2] J. Kucera, K. McKennon: Kothe's Example of an Incomplete LB-space, Proc. A.M.S. 93 (1985), 79-80. Author's address: Washington State University, Pullman, Washington 99164-2930, U.S.A. 18 

Preservation of fast completeness under inductive limits

http://dml.cz/bitstream/handle/10338.dmlcz/128301/CzechMathJ_42-1992-1_4.pdf

Preservation of fast completeness under inductive limits

Abstract

Similar works

Full text

Available Versions

Institute of Mathematics AS CR, v. v. i.