Between 2013 and 2015 Aguayo et al. developed an operator theory on the space c_0 of null sequences in the complex Levi-Civita field by defining an inner product on c_0 that induces the supremum norm on c_0 and then studying compact and self-adjoint operators on c_0, thus presenting a striking analogy between c_0 over the complex Levi-Civita field and the Hilbert space l^2 over the complex numbers field. In this thesis, we try 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 real numbers field. Two goals are considered in the thesis: (1) to present and contribute to a theory of X-normed spaces, and (2) to develop an operator theory on c_0 over a field with a Krull valuation of arbitrary rank. In order to meet the goal (1), a systematic study of valued fields, G-modules, and X-normed spaces is conducted in order to satisfy the generality of the settings required. For the goal (2), we identify the major differences between normed spaces over fields of rank 1 and X-normed spaces over fields of higher rank; and we try to find the right conditions for which the techniques employed in the rank-1 case can be used in the higher rank case. For (1) the author develops a new tool to work with transfinite induction simplifying the techniques employed in X-normed spaces, thus accomplishing a Generalized Baire Category Theorem that allows the proof of an Open Mapping theorem for X-normed spaces. Regarding (2), we show that an operator can be identified as compact with adjoint by studying the behavior of the image of any base of c_0. Although characterizations are obtained for some linear operators on c_0, it is still unknown whether the spectral theorem holds for compact self-adjoint operators in the non-Archimedean case.PFCHA-BecasPFCHA-Becas