Let $\mathbb{K}$ be a non-Archimedean, complete valued field. It is known that the supremum norm $\left\Vert \cdot\right\Vert _{\infty}$ on $c_{0}$ is induced by an inner product if and only if the residual class field of $\mathbb{K}$ is formally real. One of the main problems of this inner product is that $c_{0}$ is not orthomodular, as is any classical Hilbert space. Our goal in this work is to identify those closed subspaces of $c_{0}$ which have a normal complement. In this study we also involve projections, adjoint and self-adjoint operators.

We give a complete description of the Fredholm disjointness preserving operators between ultrametric spaces of (bounded and not necessarily bounded) continuous functions defined on ${\mathbb N}$-compact spaces.

The class of orthomodular spaces described by Gross and Künzi based on H. Keller's work is a generalization of classic Hilbert spaces. Let $E$ be an orthomodular space in this class, endowed with a positive form $\phi$. As in Hilbert spaces, $\phi$ induces a topology on $E$ making it a complete space. For every $n\in \mathbb{N}$, we describe definite spaces $(E_n,\phi_n)$, with $\dim(E_n)=2^n$ over the base field $K_n=\mathbb{R}((\chi_1,\ldots,\chi_n))$, and we build a family of selfadjoint and indecomposable operators. Later we build an orthomodular definite space $(E,\phi)$ with infinite dimension and we also prove that the sequence of operators in this family induces a bounded, selfadjoint and indecomposable operator in $(E,\phi)$.

This paper consists of a survey of the most important results on $p$-adic inductive limits obtained by the authors in recent years, together with some new results on the subject. It is mainly devoted to regularity properties and their relations with strictness and closedness properties. As a product we get the $p$-adic version of the Dieudonné-Schwartz Theorem.

The question to have an umbral calculus in non zero characteristic $p$, has been considered on some special subspaces of the space of polynomials, for instance subspaces of the space of additive polynomials with coefficients in a complete valued field of characteristic $p$ and their closure in spaces of continuous functions. In her PhD thesis, M. Héraoua has given an umbral calculus on the so called ring of formal differential operators which has a coalgebra structure. In many respects, this umbral calculus is as in the classical umbral calculus in characteristic zero. It turns out that the technique used by M. Héraoua can be extended in the topological case. More precisely, let $\mathbb{F}_q$ be the finite field with $q$ elements and $\mathbb{F}_q[[T]]$ be the ring of formal power series with coefficients in $\mathbb{F}_q$. Then with the addition of formal power series and the $T$-adic topology, $\mathbb{F}_q[[T]]$ is a totally discontinuous compact group. Let $K$ be a complete valued field, extension of the valued field of formal Laurent series $\mathbb{F}_q((T))$, then it is well known that the space of continuous functions $\mathcal{C}(\mathbb{F}_q[[T]] , K) $ is an ultrametric Hopf algebra. The coalgebra structure of $\mathcal{C}(\mathbb{F}_q[[T]] , K) $ is that of a binomial divided power coalgebra. In a previous work we have described the algebra of difference operators of the Banach coalgebra $\mathcal{C}(\mathbb{F}_q[[T]] , K) $. Here, we show that the Keigher-Pritchard's divided powers of an element of the maximal ideal of the dual algebra of $\mathcal{C}(\mathbb{F}_q[[T]] , K) $ can be performed. With this, and the fact that the algebra of difference operators is isomorphic to the latter dual algebra, one recovers much part of classical umbral calculus.

Let $K$ be an algebraically closed field, complete for a non-trivial ultrametric absolute value. We denote by $A$ the $K$- Banach algebra of bounded analytic functions in the unit disk $\{x\in K \mid \vert x\vert<1\}$. We study some properties of ideals of $A$. We show that maximal ideals of infinite codimension are not of finite type and that $A$ is not a Bezout ring.

Various definitions of $C^k$-maps on open subsets of finite-dimensional vector spaces over a complete valued field have been proposed in the literature. We show that the $C^k$-maps considered by Schikhof and De Smedt coincide with those of Bertram, Glöckner and Neeb. By contrast, Ludkovsky's $C^k$-maps need not be $C^k$ in the former sense, at least in positive characteristic. We also compare various types of Hölder differentiable maps on finite-dimensional and metrizable spaces.

In this paper, we shall study the unicity of meromorphic functions defined over non-Archimedean fields of characteristic zero such that their valence functions of poles grow slower than their characteristic functions. If $f$ is such a function, and $f$ and a linear differential polynomial $P(f)$ of $f$, whose coefficients are meromorphic functions growing slower than $f$, share one finite value $a$ CM, and share another finite value $b\ (\not=a)$ IM, then $P(f)=f$.

Let $\mathbb K$ be a complete non-Archimedean valued field and let $C(X,E)$ be the space of all continuous functions from a zero-dimensional Hausdorff topological space $X$ to a non-Archimedean Hausdorff locally convex space $E$. We will denote by $C_{b}(X,E)$ (resp. by $C_{rc}(X,E)$) the space of all $f\in C(X,E)$ for which $f(X)$ is a bounded (resp. relatively compact) subset of $E$. The dual space of $C_{rc}(X,E)$, under the topology $t_{u}$ of uniform convergence, is a space $M(X,E')$ of finitely-additive $E'$-valued measures on the algebra $K(X)$ of all clopen , i.e. both closed and open, subsets of $X$. Some subspaces of $M(X,E')$ turn out to be the duals of $C(X,E)$ or of $C_{b}(X,E)$ under certain locally convex topologies.\\ In this paper we continue with the investigation of certain subspaces of $M(X,E')$. Among other results we show that, if $E$ is a polar Fréchet space, then :\\ 1. The space $\mathcal{M}_{\theta_{o}}(X,E')$, of all $m\in M(X,E')$ for which the support of the corresponding measure $m^{\beta_{o}}$, on the Banaschewski compactification of $X$, is contained in the $\theta_{o}$-repletion of $X$, is complete under the topology of uniform convergence on the family $\mathcal{E}$ of all equicontinuous subsets $B$ of $C(X,E)$ for which $B(x)$ is a compactoid subset of $E$ for all $x\in X$.\\ 2. The space $M_{bs}(X,E')$, of all the so called strongly-separable members of $M(X,E')$ is complete under the topology of uniform convergence on the family of all uniformly bounded members of $\mathcal{E}$.\\ 3. The space $M_{s}(X,E')$ of all $m\in M(X,E')$, for which $ms$ is separable for all $s\in E$, is complete under the topology of uniform convergence on the family of all $B \in \mathcal{E}$ for which the set $B(X)$ is compactoid.

Form Hilbert spaces are constructed over fields that are complete in a non-archimedean valuation. They share with classical Hilbert spaces the basic property expressed by the Projection Theorem. However, there appear some remarkable geometric features which are unknown in Euclidean geometry. In fact, due to the so-called type condition there are only a few orthogonal straight lines containing vectors of the same length, so these non-archimedean spaces are utmost inhomogeneous. In the paper we consider a typical Form Hilbert space $(E, < , >)$ and we show that this geometric feature has a strong impact on the group $\mathcal{O}(E)$ of all isometries $T:E\longrightarrow E$ and on the lattice $\mathcal{L}$ of all normal subgroups of $\mathcal{O}$. In particular, we describe some remarkable sublattices of $\mathcal{L}$ which have no analogue in the classical orthogonal groups.

We consider a class of partial differential equations with Carlitz derivatives over a local field of positive characteristic, for which an analog of the Cauchy problem is well-posed. Equations of such type correspond to quasi-holonomic modules over the ring of differential operators with Carlitz derivatives. The above class of equations includes some equations of hypergeometric type. Building on the work of Thakur, we develop his notion of the hypergeometric function of the first kind (whose parameters belonged initially to $\mathbb Z$) in such a way that it becomes fully an object of the function field arithmetic, with the variable, parameters and values from the field of positive characteristic.

Let $K$ be a non-Archimedean, complete, densely valued field. For a given $t\in (0,1)$ we study a maximality of $t-$orthogonal sequences in $c_{0}$ over $K.$ In particular we prove that for every $t\in (0,1)$ there exists a maximal $t-$orthogonal sequence in $c_{0}$ which is not a base.

In [4], Ochsenius and Schikhof ask the following question. Given a totally ordered group $G$ with a cofinal sequence, if every element of its Dedekind completion $G^\#$ is the supremum of a sequence in $G$, does it follow that $G^\#$ is metrizable? We answer their question by studying topological properties of a family of totally ordered groups, $\Gamma_\alpha$, and their completions $\Gamma_\alpha^\#$. Furthermore we obtain for this family conditions both necessary and sufficient for the metrizability of $\Gamma_\alpha^\#$.

By means of fields of formal power series, we construct definite Banach spaces over (Krull) valued fields for any cofinal type $\geq\omega _{0}$ of the value group. Among these spaces are all known non-classical form-Hilbert spaces of infinite dimension.

Let $K$ be a non-trivially non-archimedean valued field that is complete with respect to the valuation $\left| \, \, \right| : K \longrightarrow[0,\infty)$, let $X$ be a non-empty subset of $K$ without isolated points. For $n \in \{ 0,1, \ldots \}$ the $K$-Banach space $BC^n(X)$, consisting of all $C^n$-functions $X \longrightarrow K$ whose difference quotients up to order $n$ are bounded, is defined in a natural way. It is proved that $BC^n(X)$ is of countable type if and only if $X$ is compact. In addition we will show that $BC^{\infty}(X) : = \bigcap _n BC^n(X)$, which is a Fréchet space with its usual projective topology, is of countable type if and only if $X$ is precompact.

The proof of the intermediate value theorem for power series on a Levi-Civita field will be presented. After reviewing convergence criteria for power series [19], we review their analytical properties [18]. Then we state and prove the intermediate value theorem for a large class of functions that are given locally by power series and contain all the continuations of real power series: using iteration, we construct a sequence that converges strongly to a point at which the intermediate value will be assumed.

A Fréchet space is called twisted if it is not isomorphic to a countable product of Fréchet spaces with continuous norms. It is easy to show that no non-archimedean Fréchet space with a Schauder basis is twisted. We construct examples of non-archimedean twisted nuclear Fréchet spaces.