Ultrametric Functional Analysis

This volume contains research articles based on lectures given at the Seventh International Conference on $p$-adic Functional Analysis. The articles, written by leading international experts, provide a complete overview of the latest contributions in basic functional analysis (Hilbert and Banach spaces, locally convex spaces, orthogonality, inductive limits, spaces of continuous functions, strict topologies, operator theory, automatic continuity, measure and integrations, Banach and topological algebras, summability methods, and ultrametric spaces), analytic functions (meromorphic functions, roots of rational functions, characterization of injective holomorphic functions, and Gelfand transforms in algebras of analytic functions), differential equations, Banach-Hopf algebras, Cauchy theory of Levi-Civita fields, finite differences, weighted means, $p$-adic dynamical systems, and non-Archimedean probability theory and stochastic processes.The book is written for graduate students and research mathematicians. It also would make a good reference source for those in related areas, such as classical functional analysis, complex analytic functions, probability theory, dynamical systems, orthomodular spaces, number theory, and representations of $p$-adic groups.

Table of Contents:
Non-archimedean integral operators on the space of continuous functions by J. Aguayo and M. Nova Isomorphisms with small bound between spaces of $p$-adic continuous functions by J. Araujo Automatic continuity of basis separating maps by E. Beckenstein and L. Narici Cauchy theory on Levi-Civita fields by M. Berz Uniqueness problems and applications of the ultrametric Nevanlinna theory by A. Boutabaa and A. Escassut The Hopf algebra structure of the space of continuous functions on power series over $\mathbb{F}_q$ and Carlitz polynomials by B. Diarra Metrizability of compactoid sets in non-archimedean Hausdorff (LM)-spaces by N. De Grande-de Kimpe, J. Kakol, and C. Perez-Garcia Strict topologies and vector-measures on non-archimedean spaces by A. K. Katsaras P-adic spaces with strict topologies as topological algebras by A. K. Katsaras and C. G. Petalas Non-archimedean stochastic processes by A. Khrennikov and S. Ludkovsky The asymptotic number of periodic points of discrete polynomial $p$-adic dynamical systems by A. Khrennikov, M. Nilsson, and R. Nyqvist Analysis and probability over infinite extensions of a local field, II: A multiplicative theory by A. N. Kochubei The Hahn-Banach subspaces of Banach spaces with base by A. Kubzdela On metrically universal ultrametric spaces $LV_{\tau}$ and $LW_{\tau}$ by A. J. Lemin and V. Lemin Gelfand transform and spectral radius formulae for ultrametric Banach algebras by N. Mainetti A theorem on summability factors for regular methods in complete ultrametric fields by P. N. Natarajan Hilbert-like spaces over Krull valued fields by H. Ochsenius Compact operators on non-classical Hilbert spaces by H. Ochsenius and W. H. Schikhof Locally convex spaces over non-archimedean valued fields by C. Perez-Garcia Finite-dimensional orthocomplemented subspaces in $p$-adic normed spaces by C. Perez-Garcia and W. H. Schikhof Systems of differential equations over valued fields by S. Priess-Crampe and P. Ribenboim Bi-analytic elements and partial isometries of hyperbolic space by J. Rivera-Letelier Analytic roots of solutions of $p$-adic differential equations by M.-C. Sarmant Measure theory and integration on the Levi-Civita field by K. Shamseddine and M. Berz On block basic sequences in non-archimedean Frechet spaces by W. Sliwa Dynamical systems in unramified or totally ramified extensions of the $p$-adic number field by P.-A. Svensson $p$-adic analysis and the calculus of finite differences by L. van Hamme.