Alternate Author Name(s)

Eugene M. Klimko

Document Type

Dissertation

Date of Award

1974

Keywords

Topology, Banach spaces, ultraproducts

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

First Advisor

William R. Transue

Second Advisor

David L. Hanson

Abstract

There seems to be general agreement that in this stage of the development of the theory of Banach spaces a greater wealth of examples would be useful in pointing the way for future research. There has not been a great variety of procedures for the construction of Banach spaces, but one was introduced in [3] by Bretagnolle, Dacunha-Castelle and Krivine. In [4] their procedure was presented more systematically, the space constructed was termed the ultraproduct, and various applications were made. There is also a brief discussion of ultraproducts in [7].

To our knowledge the process has not been subjected to a detailed and systematic study, and that is the purpose of this dissertation. While the definition does not require an order structure, in the applications which have been made the spaces involved have been Banach lattices and the lattice structure extends to the ultraproduct in a natural way. Hence the present-study has included these order properties.

The ultraproduct procedure has an apparent defect in that it requires free ultrafilters which cannot be explicitly constructed, but this is not as serious as might be supposed. Many properties of the ultraproduct may be obtained with merely the knowledge that the ultra-filter used contains a given filter, and the utility of the construction is well established by the applications in [3] and [4]. The property (which has been called countable intersection property) of a free ultrafilter of possessing a countable sequence of elements of the filter whose intersection is empty has shown itself to be useful. We do not know whether every free ultrafilter has this property.

In the references on ultraproducts cited above the statement of the definition of an ultraproduct of Banach spaces differs slightly from the one we have given in requiring a completion of the space as we have defined it. Since we show that the space we have defined is already complete, our definition is not in fact different from that already given.

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