Some observations on local uniform boundedness principles
Jr. Stein James D. · Czechoslovak Mathematical Journal · 1991
Although it has recently been shown that the Gliding Hump Theorem is equivalent to versions of a uniform bcundedness theorem which trace back to a result of Pták ([6]), the proof of the Gliding Hump Theorem given in ([2]) uses Ptak's idea in a very elegant fashion, and has led to speculation that the more elegant proof might lead to more powerful results.Theorem 1.Let {E n : n = 1, 2, ...} be a sequence of complete metric spaces, and letE 0 be a topological space.Let Ybe a topological space covered by a sequence {Y n : n = 1, 2, ...} of closed subsets.For n ^ 1, let R n : E n ^ E n _ i be continuous and onto.Let [T a : a eA} be a collection of mapsfrom E 0 into Y.Supposefurther that[1] for each a e A, there is an integer n such that T a R i ... R n is continuous;[2] for each x є E 0 , there is an integer n such that T a x e Y n for all a є A.Then there are integers M and N, and a non-empty open subset U of E N , such that jy^jL ... R N x є Y M for all a є A and x e U.Proof.Assume the theorem is false, and let n x = 1.Choose x t e E ni and a 1 є A such thatBy [1], we can choose an integer n 2 > п х such that the map r^! ... R ll2 is con tinuous.is open, and we can therefore choose an open set V 2 in E ni with diam V 2 < 1/2, and V 2 cz <= (T ei Ä 1 ...RJ-1 {W,) n (R2...R^)-1 (U tl ).Since the theorem has been assumed false, choose a 2 є A and x 2 є V 2 such that Т а ^г ...R" 2 x 2 ф Y 2 .Choose an open set W 2 c Y such that T a2 R t ... R n2 x 2 e W 2 and W 2 n Y 2 = 0. Since x 2 є V 2 <= (R 2 ••• Я/ 12 ) -1 (^u); we see that R 2 ... R" 2 x 2 є JJ 1V Choose an open set U l2 such that diam U 12 < 1/2, U 12 <z Ü 12 c tf lb and R 2 ••• Я" 2 х 2 e Uu-