A condition on subsets of a
Banach space E is introduced, intermediate to those of norm and linear boundedness,
which depends in an essential way on the topological as well as the linear structure of
E. It is shown that this notion, called conical boundedness, is a strictly weaker notion
than that of boundedness in some Banach spaces (including infinite dimensional
reflexive spaces and infinite dimensional Banach spaces with separable duals) and
coincides with that of boundedness in others (including l1 and all finite dimensional
spaces). After a discussion of some of the consequences of the condition of conical
boundedness and a result on general structure of convex sets in reflexive spaces in
terms of this notion, a construction is given which is valid in any nonreflexive Banach
space and which yields two characterizations of reflexive Banach spaces.
The first is in terms of (the nonexistence of) certain nonconically bounded
convex sets, and the other descibes nonreflexive spaces via the restriction
of any nonzero continuous linear functional to the unit balls of equivalent
norms.
