Inspired by seminal work, including Kaplansky's 1953 exploration of Kaplansky-Hilbert modules, along with more recent advancements in the theory of probability in vector lattices, this talk introduces a new approach to functional analysis. By substituting traditional scalars (R or C) with a (real or complex) Dedekind complete unital f-algebra L, we redefine conventional methods. The presentation will begin with a comprehensive introduction to the relevance and intrigue of this theory. Subsequent discussion will focus on our findings regarding L-Banach and L-Hilbert spaces, including a comparative analysis of their proofs with classical counterparts. This investigation is a collaborative effort with Walt van Amstel, Eder Kikianty, Miek Messerschmidt, Luan Naude, Chris Schwanke, Jan Harm van der Walt from Pretoria, and Mark Roelands from Leiden.

The seminar is open to all, aiming to broaden our perspectives and understanding of ordered structures and their applications. We encourage participants from various fields to join, contribute and expand their academic networks.