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.

We study the set of non-norm-attaining functionals on a Banach space, giving a sufficient condition for the density of this set. We also find a large class of Banach spaces for which the set of norm-attaining functionals is (dense-) lineable. In addition, among other results, we provide a new proof of the fact that every real Banach space can be equivalently renormed so that the set of non-norm-attaining functionals is non-dense.

We give a result of global Gevrey and analytic regularity on the torus for non-linear operators constructed from rigid vector fields.

We characterize the weight sequences $(M_p)_p$ such that the class of ultra-differentiable functions ${\mathcal E}_{(M_p)}$ defined by imposing conditions on the derivatives of the function in terms of this sequence coincides with a class of ultradifferentiable functions ${\mathcal E}_{(\omega)}$ defined by imposing conditions on the Fourier Laplace transform. As a corollary, we characterize the weight functions $\omega$ for which there exists a weight sequence $(M_p)_p$ such that the classes ${\mathcal E}_{(\omega)}$ and ${\mathcal E}_{(M_p)}$ coincide. These characterizations also hold in the Roumieu case. Our main results are illustrated by several examples.

In earlier research generalized multidimensional Hilbert transforms have been constructed in $\mathbb{R}^m$ in the framework of Clifford analysis, a generalization to higher dimension of the theory of holomorphic functions in the complex plane. These Hilbert transforms, obtained as part of the boundary value of an associated Cauchy transform in $\mathbb{R}^{m+1}$, might be characterized as isotropic, since the metric in the underlying space is the standard Euclidean one. In this paper we adopt the idea of a so--called anisotropic Clifford setting, leading to the introduction of a metric dependent Hilbert transform in $\mathbb{R}^m$, which formally shows similar properties as the isotropic one, but allows to adjust the co-ordinate system to preferential directions. A striking fact is that the associated Cauchy transform in $\mathbb{R}^{m+1}$ is no longer uniquely determined, but may correspond to various $(m+1)$--dimensional metrics.

The space $A(D) $ of all analytic functions in a complete $n$ -circular domain $D$ in $\mathbb{C}^{n},$ $n\geq 2,$ is considered with a natural Fréchet topology. Some sufficient conditions for the isomorphism of such spaces are obtained in terms of certain subtle geometric characteristic of domains $D$. This investigation complements essentially the second author's result on necessary geometric conditions of such isomorphisms.

We prove in this note that every separable infinite dimensional complex Fréchet space different from $\omega$, the countably infinite product of lines, admits a topologically mixing analytic uniformly continuous semigroup of operators. The study of the existence of transitive semigroups on $\omega$, and on its predual $\varphi$ is also considered.

We present a simple procedure which transfers classical coefficient tests on summation of orthonormal series in $L_2(\mu)$ into theorems on summation for unconditionally convergent series in vector-valued Banach function spaces $E(X)$ (where $E$ is a Banach function space over a measure space $(\Omega, \mu)$ and $X$ a Banach space).

Let $\mathbb{R}_{0,m}$ be the real Clifford algebra constructed over the real quadratic space $\mathbb{R}^{0,m}$ with signature $(0,m)$ and let $U_r$ be an $\mathbb{R}^+_{0,m}$-valued harmonic function in an appropriate open domain $\Omega$ of $\mathbb{R}^{m+1}$ $(0 < r \leq m; m \geq 2)$. Then a necessary and sufficient condition is given upon $U_r$ for the existence of an $\mathbb{R}^{r-1}_{0,m}$-valued harmonic function in $\Omega$ which is conjugate to $U_r$.

The paper shows that a uniform space $X$ is trans-separable if and only if every pointwise bounded uniformly equicontinuous subset of the space of continuous real-valued functions $C_{c}(X) $ equipped with the compact-open topology is metrizable. This extends earlier results of Pfister and Robertson and also applies to show that if $C_{c}(X) $ is angelic then $X$ is trans-separable. The precise relation among DCCC spaces and trans-separable spaces has been also determined.

We discuss problems of extension of vector-valued functions defined on subsets of a domain $\Omega\subset\mathbb{R}^N$ which have weak extensions belonging to a space $\mathscr{H}(\Omega)$ of smooth functions. We look for conditions which ensure that there exists an extension in the corresponding space $\mathscr{H}(\Omega,E)$ of vector-valued functions.

We give an example of a complete locally convex m-topology on the algebra of infinite differentiable functions on $[0,1]$ which is strictly coarser than the natural Fréchet-topology but finer than the topology of pointwise convergence. A similar construction works on the algebra of continuous functions on $[0,1].$ Using this examples we can separate different notions of diffotopy and homotopy.

A geometric characterization of the extension property is given for Cantor-type sets. The condition can also be done in terms of the rate of growth of certain sequences to the Robin constants of local parts of the set.

We characterize the surjective convolution operators $T_\mu$ on the space $(P_{**})'$ of Fourier ultra-hyperfunctions by means of a slowly decreasing condition for the Fourier transform $\widehat{\mu}$ and then study the existence of continuous linear right inverses for $T_\mu$.

The Kouba formula for the complex interpolation of injective tensor products requires all spaces involved to be at least of cotype $2$. We show that this can be weakened when one side of the tensor products is fixed.

Suppose that $\mathcal X$ is a family of spaces of holomorphic functions such that each $X=X(D) \in \mathcal X$ can be defined on a domain $D$ belonging to some class $\mathcal D$ of domains. Then for any two concrete domains $D_1$ and $D_2 \in \mathcal D$ and $X \in \mathcal X$ one can ask the following natural question if corresponding spaces $X(D_1)$ and $X(D_2)$ are isomorphic as topological vector spaces. Similarly, for a fixed $D \in \mathcal D$ and two different spaces $X_1, X_2 \in \mathcal X$ one can consider the existence of an isomorphism between $X_1(D)$ and $X_2(D)$. We answer these questions when $\mathcal X$ consists of Hardy $N^p_*(D)$, maximal Hardy $MN^p_*(D)$, Bergman ${\mathbb N}^p(D)$, and Lumer's Hardy $LN^p_*(D)$ algebras, $p \geq 1$, and $\mathcal D = \{\mathbb B_n, \mathbb U^n, n \in \mathbb N\}$ is the family of the unit balls and the unit polydiscs in ${C}^n$.

We introduce a new class of functions, called the $\mathcal{N}_p$-spaces and study the boundedness and compactness of composition operators on $\mathcal{N}_p$-spaces as well as between $\mathcal{N}_p$-spaces and Bergman-type spaces. The paper is intended to give a self-contained introduction the the $\mathcal{N}_p$-spaces.

In this paper, we study some Banach algebras with this property that each linear isometry between them induces a Banach algebra isometry. We obtain a Banach-Stone type theorem between Baire functions defined on completely regular spaces. As a consequence, a similar result for the space of continuous functions is deduced.

Let $(X_n)$ be a sequence of infinite-dimensional Banach spaces. For $E$ being the space $\bigoplus_{n=1}^\infty X_n$, the following equivalences are shown: 1. $E' [\mu(E',E)]$ is B-complete. 2. Every separated quotient of $E' [\mu(E',E)]$ is complete. 3. Every separated quotient of $E$ satisfies Mackey's weak condition. 4. $X_n$ is quasi-reflexive, $n\in \mathbb{n}$.

It is shown that for a wide class of linear partial differential operators with constant coefficients the space of real analytic zero solutions does not admit a Schauder basis. This is based on results on the linear topological structure of the space of zero solutions and a careful analysis of the solvability with a real analytic parameter.

A characterization in terms of the weights of quasinormable weighted Fréchet spaces of holomorphic functions on the disc is presented, when the weights are radial and grow logarithmically.