L-functional analysis

Inspired by the theories of Kaplansky-Hilbert modules and probability theory in vector lattices, we generalise functional analysis by replacing the scalars R or C by a real or complex Dedekind complete unital f-algebra L; such an algebra can be represented as a suitable space of continuous functions. We set up the basic theory of L-normed and L-Banach spaces and bounded operators between them, we discuss the L-valued analogues of the classical ᵖ-spaces, and we prove the analogue of the Hahn-Banach theorem. We also discuss the basics of the theory of L-Hilbert spaces, including projections onto convex subsets and the Riesz Representation theorem.