International Journal of Mathematics and Mathematical Sciences
Volume 32 (2002), Issue 3, Pages 149-165

Boundedness and surjectivity in normed spaces

Olav Nygaard

Department of Mathematics, Agder University College, Servicebox 422, Kristiansand 4604, Norway

Received 24 January 2001; Revised 6 June 2001

Copyright © 2002 Olav Nygaard. 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.


We define the (w* -) boundedness property and the (w* -) surjectivity property for sets in normed spaces. We show that these properties are pairwise equivalent in complete normed spaces by characterizing them in terms of a category-like property called (w* -) thickness. We give examples of interesting sets having or not having these properties. In particular, we prove that the tensor product of two w*-thick sets in X** and Y* is a w*-thick subset in L(X,Y)* and obtain as a consequence that the set w*-expBK(l2)* is w*-thick.