  Using the bicomplex numbers $\mathbb{T}$ which is a commutative ring with
zero divisors defined by $\mathbb{T}=\{w_0 + w_1 i_1 + w_2 i_2 + w_3 j | w_0,
w_1, w_2, w_3 \in \mathbb{R}\}$ where $i_{1}^{2} = -1, i_{2}^{2} = -1, j^2 = 1,
i_1 i_2 = j = i_2 i_1$, we construct hyperbolic and bicomplex Hilbert spaces.
Linear functionals and dual spaces are considered and properties of linear
operators are obtained; in particular it is established that the eigenvalues of
a bicomplex self-adjoint operator are in the set of hyperbolic numbers.
