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A family $\{A_{\alpha }:\alpha \in \mathbb{N}^{\mathbb{N}}\}$ of sets covering a set $E$ is called a resolution for $E$ if $A_{\alpha }\subseteq A_{\beta }$ whenever $\alpha \leq \beta $. A locally convex space (lcs) $E$ is said to belong to class $\mathfrak{G}$ if there is a resolution $\{A_{\alpha }:\alpha \in \mathbb{N}^{\mathbb{N}}\}$ for $(E^{\prime},\sigma (E^{\prime },E))$ such that each sequence in any $A_{\alpha }$ is equicontinuous. The class $\mathfrak{G}$ contains `almost all' useful locally convex spaces (including $(LF)$-spaces and $(DF)$-spaces). We show that$\;\left( i\right) $ every semi-reflexive lcs $E$ in class $\mathfrak{G}$ is a Lindelof $\Sigma $-space in the weak topology (this extends a corresponding result of Preiss-Talagrand for WCG Banach spaces) and the weak* dual of $E$ is both $K$-analytic and has countable tightness, $\left( ii\right) $ a barrelled space $E$ has a weakly compact resolution if and only if $E$ is weakly $K$-analytic, and $\left( iii\right) $ if $E$ is barrelled or bornological then $E^{\prime }$ has a weak* compact resolution if and only if it is weak* $K$-analytic. As an additional consequence we provide another approach to show that the weak* dual of a quasi-barrelled space in class $\mathfrak{G}$ is $K$-analytic. These results supplement earlier work of Talagrand, Preiss, Cascales, Ferrando, Kąkol, Lopez Pellicer and Saxon.

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.

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.

In the introductory Section 0. some of the most important points in the professional curriculum vitae of Jean Schmets are given. Then Section 1 is devoted to present (part of) the work for which Schmets was very well known until 1990: He has been the leading specialist in the world for locally convex spaces $C(X)$ of continuous functions with various topologies and for the corresponding spaces $C(X,E)$ of vector valued continuous functions. Finally, in Section 2 some results which Schmets obtained in cooperation with Manuel Valdivia since 1990 are reviewed: work on domains of real analytic existence, continuous linear right inverses for $C^\infty$- functions and Whitney extensions for non quasianalytic functions.

In this short note we prove that the locally convex inductive limit of commutative locally m-convex algebras is already locally m-convex whenever multiplication is continuous. A topological algebra is an (associative) algebra endowed with a vector space topology such that multiplication is jointly continuous. If one considers inductive limits (in the category of topological vector spaces) of topological algebras multiplication may fail to be continuous since it is a bilinear map on the product which is certainly not linear, and forming inductive limits only respects the linear structure (although this is quite clear, the dierence between linearity and bilinearity in this context has been overlooked several times in the literature). The most important category of topological algebras is that of locally m-convex (l.m.c.) algebras having a 0-neighbourhood basis consisting of m-convex sets (i.e. absolutely convex sets B which are multiplicative in the sense that B 2 B). This class was introduced by Michael [9] and Arens [2] since it allows generalizations and applications of the theory of Banach algebras. Since their work a number of authors studied the question whether the inductive limit (in the category of locally convex spaces) of l.m.c. algebras is again l.m.c., see e.g. [1, 3, 4, 5, 7, 8, 10]. This is indeed true for countable inductive limits of normed algebras [1, 4]. On the other hand, Warner [10] gave an example of an inductive limit of metrizable l.m.c. algebras which is not l.m.c., and in [4] it is shown that [

Let $Q$ be a bounded, convex and locally closed subset of \ $\C^N$, let $H(Q)$ be the space of all functions which are holomorphic on an open neighborhood of $Q$. We endow $H(Q)$ with its projective topology. We show that the topology of the weighted inductive limit of Fr\'echet spaces of entire functions which is obtained as the Laplace transform of the strong dual to $H(Q)$ can be described be means of canonical weighted seminorms if and only if the intersection of $Q$ with each supporting hyperplane to the closure of $Q$ is compact. We also find conditions under which this (LF)-space of entire functions coincides algebraically with its projective hull.