By Jesus Araujo-gomez, Bertin Diarra, Alain Escassut
This quantity includes papers according to lectures given on the 11th overseas convention on $p$-adic practical research, which was once held from July 5-9, 2010, in Clermont-Ferrand, France. The articles accumulated right here characteristic fresh advancements in quite a few parts of non-Archimedean research: Hilbert and Banach areas, finite dimensional areas, topological vector areas and operator idea, strict topologies, areas of constant capabilities and of strictly differentiable capabilities, isomorphisms among Banach features areas, and degree and integration. different issues mentioned during this quantity comprise $p$-adic differential and $q$-difference equations, rational and non-Archimedean analytic features, the spectrum of a few algebras of analytic services, and maximal beliefs of the ultrametric corona algebra
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Extra info for Advances in non-Archimedean Analysis: 11th International Conference P-adic Functional Analysis July 5-9, 2010 Universite Blaise Pascal, Clermont-ferrand, France
We may also assume that the Ai ’s are polynomials. , As (X) (s ≥ 1) are polynomials of K[X] such that A0 (X)As (X) = 0. 2 in  and will be useful for the proof of Theorem 2: Proposition 4: Let f be a meromorphic solution in K of Equation (E”). Then, if α is a non zero pole of f , there exists an integer m ∈ N and a zero θ of A0 diﬀerent from zero such that α = q −m θ and ωθ (A0 ) + ωα (f ) ≥ 0. Particularly, every non zero pole α of satisﬁes −ωα (f ) ≤ deg A0 . Proof: As, in the proof of Theorem 1, we may suppose that f has no pole at 0.
Strictness of p-Adic inductive sequences was one of the favourite subjects of Nicole. (En )n is called strict if τn+1 |En = τn for all n. 24 C. H. SCHIKHOF In  Nicole and C. Perez-Garcia applied the results of  to get relations between several notions of strictness and some closedness properties. The construction of certain classes of inductive sequences carried out in the last section of  (a “machine” to construct examples of inductive sequences, as Nicole put it) provided most of the examples needed along that paper.
Ix) E and Eb are strictly of countable type, Montel (hence reﬂexive) and nuclear. In particular, E is the strong dual of the Fr´echet space Eb . 16. Since E is strictly of countable type (see (ix)), it is polar. 13, for compactoid inductive sequences of polar Banach spaces, the “ordinary” and the polar limits coincide. 12 are compactoid. The answer was given in  as follows. I: this inductive sequence is strict and, since its steps are inﬁnitedimensional Banach spaces, it cannot be compactoid.