International Journal of Mathematics and Mathematical Sciences
Volume 8 (1985), Issue 1, Pages 189-191

Two large subsets of a functional space

F. S. Cater

Department of Mathematics, Portland State University, Portland 97207, Oregon, USA

Received 29 May 1984

Copyright © 1985 F. S. Cater. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let P1 denote the Banach space composed of all bounded derivatives f of everywhere differentiable functions on [0,1] such that the set of points where f vanishes is dense in [0,1]. Let D0 consist of those functions in P that are unsigned on every interval, and let D1 consist of those functions in P1 that vanish on dense subsets of measure zero. Then D0 and D1 are dense Gδ-subsets of P1 with void interior. Neither D0 nor D1 is a subset of the other.