Spaces of real valued functions form many examples in the theory of ordered vector spaces and vector lattices. For example the space of real -valued polynomials on a bounded subset of the reals is not in general a vector lattice but does have the Riesz interpolation property (RIP). Another example of an ordered vector space which is in general not a lattice is the space of differentiable functions, again this space has the Riesz decomposition property (RDP) which is equivalent to the RIP for ordered vector spaces. In this thesis we replace the real -valued function versions of these spaces with Banach lattice-valued functions with appropriate definitions and investigate when they have the RIP/RDP. We also consider when the Banach lattice-valued polynomials form a vector lattice. Spaces of continuous real-valued functions on a (locally) compact Hausdorff space are very important in Banach lattice theory. Generalisations to Banach lattice-valued functions have already been made and many analogous results to the real case have been proved. Including some extension and separation results. In the thesis this space is further generalised by considering functions which are continuous with respect to the weak topology on the Banach lattice instead of the norm topology. These spaces of functions may not be Banach lattices in general in contrast to the norm continuous versions which always are. We provide conditions under which it will be a Banach lattice and prove several other desirable properties for a Banach lattice under varying conditions. We also give some extension and separtion results under fairly restrictive conditions.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:675660 |
| Date | January 2014 |
| Creators | McKenzie, Stuart Lamont James |
| Publisher | Queen's University Belfast |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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