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