Pinchuck, Andrew L.2011-06-062011-06-062011-06-06http://hdl.handle.net/10539/10006We prove an original version of the Hahn-Banach theorem in the fuzzy setting. Convex compact sets occur naturally in set-valued analysis. A question that has not been satisfactorily dealt with in the literature is: What is the relationship between collections of such sets and vector spaces? We thoroughly clarify this situation by making use of R°adstr ¨om’s embedding theorem, leading up to the definition of a near vector space. We then go on to successfully apply these results to provide an original method of proof of Doob’s decomposition of submartingales.enThesis