Let $\mathcal {N}$ be a non-Archimedean-ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order, and whose Hahn group is Archimedean. In this paper, we first review the properties of weakly locally uniformly differentiable (WLUD) functions, $k$ times weakly locally uniformly differentiable (WLUD$^{k}$) functions and WLUD$^{\infty }$ functions at a point or on an open subset of $\mathcal {N}$. Then, we show under what conditions a WLUD$^{\infty }$ function at a point $x_0\in \mathcal {N}$ is analytic in an interval around $x_0$, that is, it has a convergent Taylor series at any point in that interval. We generalize the concepts of WLUD$^{k}$ and WLUD$^{\infty }$ to functions from $\mathcal {N}^{n}$ to $\mathcal {N}$, for any $n\in \mathbb {N}$. Then, we formulate conditions under which a WLUD$^{\infty }$ function at a point $\boldsymbol {x_0} \in \mathcal {N}^{n}$ is analytic at that point.

This work focuses on the ongoing research of lineability (the search for large linear structures within certain non-linear sets) in non-Archimedean frameworks. Among several other results, we show that there exist large linear structures inside each of the following sets: (i) functions with a fixed closed subset of continuity, (ii) all continuous functions that are not Darboux continuous (or vice versa), (iii) all functions whose Dieudonné integral does not behave as an antiderivative, and (iv) functions with finite range and having antiderivative.

We define and study generalizations of simplicial volume over arbitrary seminormed rings with a focus on p-adic simplicial volumes. We investigate the dependence on the prime and establish homology bounds in terms of p-adic simplicial volumes. As the main examples, we compute the weightless and p-adic simplicial volumes of surfaces. This is based on an alternative way to calculate classical simplicial volume of surfaces without hyperbolic straightening and shows that surfaces satisfy mod p and p-adic approximation of simplicial volume.

Using convex subrings of *$\mathbb{C}$, a nonstandard extension of $\mathbb{C}$, we define several kinds of complex bounded polynomials and we provide their associated analytic functions obtained by taking the quasistandard part.

The study carried out in this paper about some new examples of Banach spaces, consisting of certain valued fields extensions, is a typical non-archimedean feature. We determine whether these extensions are of countable type, have $t$-orthogonal bases, or are reflexive. As an application we construct, for a class of base fields, a norm $\left\| \,.\, \right\|$ on ${{c}_{0}}$, equivalent to the canonical supremum norm, without non-zero vectors that are $\left\| \,.\, \right\|$-orthogonal and such that there is a multiplication on ${{c}_{0}}$ making $\left( {{c}_{0}},\,\left\| \,.\, \right\| \right)$ into a valued field.

The non-archimedean power series spaces, ${{A}_{1}}(a)\,\text{and}\,{{A}_{\infty }}(b)$, are the best known and most important examples of non-archimedean nuclear Fréchet spaces. We prove that the range of every continuous linear map from ${{A}_{p}}(a)\,\text{to}\,{{A}_{q}}(b)$ has a Schauder basis if either $p\,=\,1$ or $p\,=\,\infty $ and the set ${{M}_{b,a}}$ of all bounded limit points of the double sequence (${{({{b}_{i}}/{{a}_{j}})}_{i,j\in \mathbb{N}}}$ is bounded. It follows that every complemented subspace of a power series space ${{A}_{p}}(a)$ has a Schauder basis if either $p\,=\,1$ or $p\,=\,\infty $ and the set ${{M}_{a,a}}$ is bounded.

It is proved that every infinite-dimensional non-archimedean Banach space of countable type admits a linear continuous operator without a non-trivial closed invariant subspace. This solves a problem stated by A. C. M. van Rooij and W. H. Schikhof in 1992.

The General Curve Lemma is a tool of Infinite-Dimensional Analysis that enables refined studies of differentiability properties of maps between real locally convex spaces to be made. In this article, we generalize the General Curve Lemma in two ways. First, we remove the condition of local convexity in the real case. Second, we adapt the lemma to the case of curves in topological vector spaces over ultrametric fields.

We study sets of range uniqueness (SRUs) for analytic functions inside a disc of an algebraically closed field $K$ complete with respect to an ultrametric absolute value. The SRUs we obtain are converging sequences. We first obtain results that look like those known in $\mathbb{C}$ but involve a weaker hypothesis than in $\mathbb{C}$: let $(a_n)$ be a sequence of limit $a$ in a disc $d(a,r^-)$ such that $|a_n-a|$ is a strictly decreasing sequence. If the sequence $(a_n)$ does not make an SRU for the set $\mathcal{A}(d(a,r^-))$ of analytic functions inside $d(a,r^-)$, then, for a certain integer $k\in\mathbb{Z}$, the sequence

$$ \bigg(\frac{a_{n+k}-a}{a_n-a}\bigg) $$

has a finite limit in $K$ and the sequence

$$ \bigg(\frac{\log|a_{n+k}-a|}{\log|a_n-a|}\bigg) $$

has a finite rational limit. Next, we show that if the sequence

$$ \frac{\log(a_{n+1}-a)}{\log(a_n-a)} $$

converges to a limit $b\geq1$ in such a way that $-b\log|a_{n}-a|\lt-b\log|a_{n+1}-a|$ and if $\log|a_{n}-a|-b\log|a_{n+1}-a|$ has limit $0$ or $+\infty$ and if $b^k\notin\mathbb{Q}$ whenever $b>1$ and $k\in \mathbb{N}^*$, then the sequence $(a_n)$ is an SRU for $\mathcal{A}(d(a,r^-))$. In particular, for every $\gamma\in\;]0,1[\;\cup\;]1,+\infty[$, $L\in\mathbb{Q}\;\cap\;]0,+\infty[$ and $b\geq 1$, there exist SRUs for $\mathcal{A}(d(a,r^-))$ of the form $\{a_n\mid n\in\mathbb{N}\}$ such that

$$ \lim_{n\rightarrow+\infty}\frac{-\log|a_n-a|}{b^nn^\gamma}=L. $$

For example, if $\gamma\in\mathbb{N}$ with $\gamma\neq0,1$, there exist SRUs of the form $\{a_n\mid n\in\mathbb{N}\}$ such that $-\log |a_n-a|=Ln^\gamma$ for all $n\in\mathbb{N}^*$. The latter result ceases to hold when $\gamma=1$. Many examples and counterexamples are provided.

We show that there exists a non-archimedean Fréchet-Montel space $W$ with a basis and with a continuous norm such that any non-archimedean Fréchet space of countable type is isomorphic to a quotient of $W$. We also prove that any non-archimedean nuclear Fréchet space is isomorphic to a quotient of some non-archimedean nuclear Fréchet space with a basis and with a continuous norm.

Let K be an algebraically closed field of characteristic zero, complete with respect to an ultrametric absolute value. In a previous paper, we had found URSCM of 7 points for the whole set of unbounded analytic functions inside an open disk. Here we show the existence of URSCM of 5 points for the same set of functions. We notice a characterization of BI-URSCM of 4 points (and infinity) for meromorphic functions in K and can find BI-URSCM for unbounded meromorphic functions with 9 points (and infinity). The method is based on the p-Adic Nevanlinna Second Main Theorem on 3 Small Functions applied to unbounded analytic and meromorphic functions inside an open disk and we show a more general result based upon the hypothesis of a finite symmetric difference on sets of zeros, counting multiplicities.

It is known that any non-archimedean Fréchet space of countable type is isomorphic to a subspace of $c_{0}^{\mathbb{N}}$. In this paper we prove that there exists a non-archimedean Fréchet space $U$ with a basis $({{u}_{n}})$ such that any basis $({{x}_{n}})$ in a non-archimedean Fréchet space $X$ is equivalent to a subbasis $({{u}_{kn}})$ of $({{u}_{n}})$. Then any non-archimedean Fréchet space with a basis is isomorphic to a complemented subspace of $U$. In contrast to this, we show that a non-archimedean Fréchet space $X$ with a basis $({{x}_{n}})$ is isomorphic to a complemented subspace of $c_{0}^{\mathbb{N}}$ if and only if $X$ is isomorphic to one of the following spaces: ${{c}_{0}},\,{{c}_{0}}\,\times \,{{\mathbb{K}}^{\mathbb{N}}},\,{{\mathbb{K}}^{\mathbb{N}}},\,c_{0}^{\mathbb{N}}$. Finally, we prove that there is no nuclear non-archimedean Fréchet space $H$ with a basis $({{h}_{n}})$ such that any basis $({{y}_{n}})$ in a nuclear non-archimedean Fréchet space $Y$ is equivalent to a subbasis $({{h}_{kn}})$ of $({{h}_{n}})$.

For a subset $\Lambda$ of the dual group of a compact metrizable abelian group, we introduce the type I-, II-, and III-$\Lambda$-Riemann–Lebesgue property of a Banach space. As an application we use these properties to characterize Rajchman sets.

For finite-dimensional subspaces of ℓ∞ over a non-archimedean valued base field K we study orthocomplementation as related to the Hahn-Banach property and strictness. As a corollary we obtain that, if K is not spherically complete, a closed hyperplane in c0 having the Hahn-Banach property is orthocomplemented (Theorem 2.1, Remark 2).

We generalize the notion of t-orthogonality in p-adic Banach spaces by introducing t-frames (§2). This we use to prove that a Fréchet-Montel (FM-)space is of countable type (Theorem 3.1), the non-archimedeancounterpart of a well known theorem in functional analysis over ℝ or ℂ ([6], p. 231). We obtain several characterizations of FM-spaces (Theorem 3.3) and characterize the nuclear spaces among them (§4).