Barría Comicheo, Angel2018-11-302018-11-302018-11-282018-11-30Barria Comicheo, A., and Shamseddine, K. Summary on non-Archimedean valued fields. In Advances in Ultrametric Analysis (2018), A. Escassut, C. Perez-Garcia, and K. Shamseddine, Eds., vol. 704 of Contemporary Mathematics, Providence, RI: American Mathematical Society, pp. 1–36.http://hdl.handle.net/1993/33572Between 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.engnon-Archimedean Functional AnalysisOperator theoryNon-Archimedean valued fieldsdoctoral thesis