Vol. 44, No. 2, 1973

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ISSN: 0030-8730
Conically bounded sets in Banach spaces

Richard David Bourgin

Vol. 44 (1973), No. 2, 411–419

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.

Mathematical Subject Classification
Primary: 46B05
Received: 18 October 1971
Revised: 15 March 1972
Published: 1 February 1973
Richard David Bourgin