Modules over semifields

The content of this book touches on algebra, topology and functional analysis. Ring modules, which are generalizations of vector spaces, are studied. In this respect, ring structures are choosen, which are with regard to their properties, close to real numbers. Thus, the aim ist to generalize propositions for these ring modules that belong to the stock of the theory of vector spaces, functional analysis or the topology, what may introduce a broader view to the analysis.
For that purpose, topological semifields are introduced and examined in the context of complete vector lattices. Furthermore, convexity in semifield modules is defined and studied that is the basis for the generalization of a classic result by A.N. Kolmogorov concerning the normability of vector space topologies.
In 1935,Jordan and von Neumann found the parallelogram equation as a characterization of the normed vector spaces whose norm can be generated by an inner product. This theorem is transferred to semifield modules.
Extension theorems are central statements of the functional analysis with various consequences. Such propositions are developed for monotonous semifield-valued functionals defined on semifield modules, from which propositions of the Hahn-Banach type can be derived. The latter are proven directly too.
Finally, extension theorems are applied to the proofs of existence theorems for semifield-valued functionals.