Between 2013 and 2015 Aguayo et al. developed an operator theory on the space c0 of null sequences in the complex Levi-Civita field by defining an inner product on c0 that induces the supremum norm on c0 and then studying compact and self-adjoint operators on c0, thus presenting a striking analogy between c0 over the complex Levi-Civita field and the Hilbert space l2 over the complex numbers field. In this thesis, the author tries to obtain these results in the most general case possible by considering a base field with a Krull valuation taking values in an arbitrary commutative group. This leads to the concept of X-normed spaces, which are spaces with norms taking values in a totally ordered set X not necessarily embedded in the field of real numbers.