Boundedness and Surjectivity in Normed Spaces

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 concequense that the set w∗ − exp BK(l2)∗ is w ∗-thick. Introduction Recall the Banach-Steinhaus theorem for Banach spaces: A family of linear continuous operators on a Banach space X, which is pointwise bounded on a set of second category, is bounded. Let X be a normed linear space. Motivated by the Banach-Steinhaus theorem we say that A ⊂ X has the boundedness property if every family of linear continuous operators on X, which is pointwise bounded on A, is bounded. More generally, if Y is a normed space and A is a subset of L(X,Y ), we say that A has the Arestricted boundedness property if every family of linear continuous operators in A, which is pointwise bounded on A, is bounded. In the latter definition, if A is the space of adjoints, we say that A ⊂ X has the w-boundedness property . From the proof of the Banach-Steinhaus theorem we conclude that every second category set A in a Banach space X has the boundedness property. However, the Nikodym-Grothendieck boundedness theorem (see e.g. [3, p. 14] or [2, p. 80]) says in our terminology exactly that the set of characteristic functions on the unit sphere of B(Σ) has the boundedness property. This set is certainly not of the second category, it is even nowhere dense. Thus it may be possible to sharpen the Banach-Steinhaus theorem. Let us have a look at a more recent theorem of J. Fernandez [5] (see also [23]), which is in the same spirit as the Nikodym-Grothendieck theorem. Date: February 1, 2008. 1991 Mathematics Subject Classification. Primary: 46B20, 46B25, 28A33; Secondary: 30H05.