  With the goal of providing the foundations for a rigorous study of modules of
bicomplex holomorphic functions, we develop a general theory of functional
analysis with bicomplex scalars. Even though the basic properties of bicomplex
number are well known and widely available, our analysis requires some more
delicate discussion of the various structures which are hidden in the ring of
bicomplex numbers. We study in particular matrices with bicomplex numbers,
bicomplex modules, and inner products and norms in bicomplex modules. We
consider two kinds of norms on bicomplex modules: a real-valued norm (as one
would expect), and a hyperbolic-valued norm. Interestingly enough, while both
norms can be used to build the theory of normed bicomplex modules, the
hyperbolic-valued norm appears to be much better compatible with the structure
of bicomplex modules. We also consider linear functionals on bicomplex numbers.
To conclude we describe a bicomplex version of classical Schur analysis
