  This paper begins the study of infinite-dimensional modules defined on
bicomplex numbers. It generalizes a number of results obtained with
finite-dimensional bicomplex modules. The central concept introduced is the one
of a bicomplex Hilbert space. Properties of such spaces are obtained through
properties of several of their subsets which have the structure of genuine
Hilbert spaces. In particular, we derive the Riesz representation theorem for
bicomplex continuous linear functionals and a general version of the bicomplex
Schwarz inequality. Applications to concepts relevant to quantum mechanics,
specifically the bicomplex analogue of the quantum harmonic oscillator, are
pointed out.
