It is proved that the free topological vector space $\mathbb{V}([0,1])$ contains an isomorphic copy of the free topological vector space $\mathbb{V}([0,1]^{n})$ for every finite-dimensional cube $[0,1]^{n}$, thereby answering an open question in the literature. We show that this result cannot be extended from the closed unit interval $[0,1]$ to general metrisable spaces. Indeed, we prove that the free topological vector space $\mathbb{V}(X)$ does not even have a vector subspace isomorphic as a topological vector space to $\mathbb{V}(X\oplus X)$, where $X$ is a Cook continuum, which is a one-dimensional compact metric space. This is also shown to be the case for a rigid Bernstein set, which is a zero-dimensional subspace of the real line.

The algebra of all Dirichlet series that are uniformly convergent in the half-plane of complex numbers with positive real part is investigated. When it is endowed with its natural locally convex topology, it is a non-nuclear Fréchet Schwartz space with basis. Moreover, it is a locally multiplicative algebra but not a Q-algebra. Composition operators on this space are also studied.

Let $G$ be a locally compact group with a fixed left Haar measure. In this paper, given a strictly positive Young function ${\rm\Phi}$, we consider $L^{{\rm\Phi}}(G)$ as a Banach left $L^{1}(G)$-module. Then we equip $L^{{\rm\Phi}}(G)$ with the strict topology induced by $L^{1}(G)$ in the sense of Sentilles and Taylor. Some properties of this locally convex topology and a comparison with weak$^{\ast }$, bounded weak$^{\ast }$ and norm topologies are presented.

Let $L\left( X \right)$ be the free locally convex space over a Tychonoff space $X$. Then $L\left( X \right)$ is a $k$-space if and only if $X$ is a countable discrete space. We prove also that $L\left( D \right)$ has uncountable tightness for every uncountable discrete space $D$.

For a Lie group $G$, we show that the map $C_{c}^{\infty }\,\left( G \right)\,\times \,C_{c}^{\infty }\,\left( G \right)\,\to \,C_{c}^{\infty }\,\left( G \right),\,\left( \gamma ,\,\eta \right)\mapsto \,\gamma \,*\,\eta $ taking a pair of test functions to their convolution, is continuous if and only if $G$ is $\sigma -$compact. More generally, consider $r,\,s,\,t\,\in {{\mathbb{N}}_{0}}\,\cup \,\left\{ \infty \right\}$ with $t\,\le \,r\,+\,s$, locally convex spaces ${{E}_{1}}\,,\,{{E}_{2}}$ and a continuous bilinear map $b:\,{{E}_{1}}\,\times \,{{E}_{2}}\,\to \,F$ to a complete locally convex space $F$. Let $\beta :\,C_{c}^{r}\,\left( G,\,{{E}_{1}} \right)\,\times \,C_{c}^{S}\,\left( G,\,{{E}_{2}} \right)\,\to$$C_{c}^{t}\,\left( G,\,F \right),\,\left( \gamma ,\,\eta \right)\,\mapsto \,\gamma \,*\,b\,\eta$ be the associated convolution map. The main result is a characterization of those $\left( G,\,r,s,t,b \right)$ for which $\beta$ is continuous. Convolution of compactly supported continuous functions on a locally compact group is also discussed as well as convolution of compactly supported ${{L}^{1}}$-functions and convolution of compactly supported Radon measures.

For a locally compact group $ \mathcal{G} $, we introduce and study a class of locally convex topologies $\tau $ on the measure algebra $M( \mathcal{G} )$ of $ \mathcal{G} $. In particular, we show that the strong dual of $(M( \mathcal{G} ), \tau )$ can be identified with a closed subspace of the Banach space $M\mathop{( \mathcal{G} )}\nolimits ^{\ast } $; we also investigate some properties of the locally convex space $(M( \mathcal{G} ), \tau )$.

Answering a question of W. Arendt and M. Kunze in the negative, we construct a Banach space X and a norm closed weak* dense subspace Y of the dual X′ of X such that X, endowed with the Mackey topology μ(X,Y ) of the dual pair 〈X,Y 〉, is not complete.

It is shown that the dual of the space Cp(I) of all real-valued continuous functions on the closed unit interval with the pointwise topology, when equipped with the Mackey topology, is a non K-analytic but weakly analytic locally convex space.

Solutions are provided to several questions concerning topologically transitive and hypercyclic continuous linear operators on Hausdorff locally convex spaces that are not Fréchet spaces. Among others, the following results are presented. (1) There exist transitive operators on the space $\varphi $ of all finite sequences endowed with the finest locally convex topology (it was already known that there is no hypercyclic operator on $\varphi$). (2) The space of all test functions for distributions, which is also a complete direct sum of Fréchet spaces, admits hypercyclic operators. (3) Every separable infinite-dimensional Fréchet space contains a dense hyperplane that admits no transitive operator.

We prove Hahn-Banach type theorems for linear functionals with values in R∪{+∞} on ordered cones, Using the concept of locally convex cones, we provide a sandwich theorem involving sub- and superlinear functionals which are allowed to attain infinite values. It render general versions of well-known extension and separation results. We describe the range of all linear functionals sandwiched between given sub- and superlinear functionals on an ordered cone. The results are of interest even in vector spaces, since we consider sublinear functionals that may attain the value +∞.