Non-Archimedean Analysis A Systematic Approach to Rigid Analytic Geometry [Paperback]

A. Linear Ultrametric Analysis and Valuation Theory.- 1. Norms and Valuations.- 1.1. Semi-normed and normed groups.- 1.1.1. Ultrametric functions.- 1.1.2. Filtrations.- 1.1.3. Semi-normed and normed groups. Ultrametric topology.- 1.1.4. Distance.- 1.1.5. Strictly closed subgroups.- 1.1.6. Quotient groups.- 1.1.7. Completions.- 1.1.8. Convergent series.- 1.1.9. Strict homomorphisms and completions.- 1.2. Semi-normed and normed rings.- 1.2.1. Semi-normed and normed rings.- 1.2.2. Power-multiplicative and multiplicative elements.- 1.2.3. The category 𝔑 and the functor A ? A~.- 1.2.4. Topologically nilpotent elements and complete normed rings.- 1.2.5. Power-bounded elements.- 1.3. Power-multiplicative semi-norms.- 1.3.1. Definition and elementary properties.- 1.3.2. Smoothing procedures for semi-norms.- 1.3.3. Standard examples of norms and semi-norms.- 1.4. Strictly convergent power series.- 1.4.1. Definition and structure of A?X?.- 1.4.2. Structure of A?X??.- 1.4.3. Bounded homomorphisms of A?X?.- 1.5. Non-Archimedean valuations.- 1.5.1. Valued rings.- 1.5.2. Examples.- 1.5.3. The Gauss-Lemma.- 1.5.4. Spectral value of monic polynomials.- 1.5.5. Formal power series in countably many indeterminates.- 1.6. Discrete valuation rings.- 1.6.1. Definition. Elementary properties.- 1.6.2. The example of F. K. Schmidt.- 1.7. Bald and discrete B-rings.- 1.7.1. B-rings.- 1.7.2. Bald rings.- 1.8. Quasi-Noetherian B-rings.- 1.8.1. Definition and characterization.- 1.8.2. Construction of quasi-Noetherian rings.- 2. Normed modules and normed vector spaces.- 2.1. Normed and faithfully normed modules.- 2.1.1. Definition.- 2.1.2. Submodules and quotient modules.- 2.1.3. Modules of fractions. Completions.- 2.1.4. Ramification index.- 2.1.5. Direct sum. Bounded and restricted direct product.- 2.1.6. The module L(L, M) of bounded A-linear maps.- 2.1.7. Complete tensor products.- 2.1.8. Continuity and boundedness.- 2.1.9. Density condition.- 2.1.10. The functor M ? M~. Residue dl“