We introduce the concept of *-mapping as a selection of the duality mapping. We prove that the *-mappings are more general than the support mappings and provide a simple proof of the characterisation of smoothness by the norm to weak-star continuity of the *-mappings. As a consequence, we provide a characterisation of Hilbert spaces in terms of *-mappings and show that a Banach space has the Schur property if and only if it has the Dunford-Pettis property and there exists a *-mapping that is sequentially w – w continuous at 0. This last fact leads to the existence of smooth Banach spaces on which the duality mapping is not sequentially w – w continuous at 0.
The Mathematical Proceedings of the Royal Irish Academy is a peer-reviewed journal which publishes original research in pure and applied mathematics. Exceptionally, survey articles of topics of current research interest that present new points of view or major simplifications are also published.
The Royal Irish Academy, the academy for the sciences and humanities for the whole of Ireland will vigorously promote excellence in scholarship, recognise achievements in learning, direct research programmes and undertake its own research projects, particularly in areas relating to Ireland and its heritage. It will reflect upon, advise on and contribute to public debate and public policy formation on issues of major interest in science, technology and culture. It will continue to offer an independent forum to Irish scholars, it will provide a network of support for scholarly disciplines through its network of national committees and commissions, it will maintain and enhance its unique library, it will publish scholarly papers and it will represent the world of Irish learning internationally.
This item is part of a JSTOR Collection.
For terms and use, please refer to our Terms and Conditions
© Royal Irish Academy
Request Permissions
