We prove that any infinite-dimensional non-archimedean Fréchet space $E$ is homeomorphic to $D^{\mathbb{N}}$ where $D$ is a discrete space with $\mathrm{c}ard\left(D\right)=\mathrm{d}ens\left(E\right)$. It follows that infinite-dimensional non-archimedean Fréchet spaces $E$ and $F$ are homeomorphic if and only if $\mathrm{d}ens\left(E\right)=\mathrm{d}ens\left(F\right)$. In particular, any infinite-dimensional non-archimedean Fréchet space of countable type over a field $𝕂$ is homeomorphic to the non-archimedean Fréchet space ${𝕂}^{\mathbb{N}}$.

A (Hausdorff) topological group is said to have a -base if it admits a base of neighbourhoods of the unit, $U_{\alpha }:\alpha \in {\mathbb{N}}^{\mathbb{N}}$, such that $U_{\alpha }\subset U_{\beta }$ whenever β ≤ α for all $\alpha ,\beta \in {\mathbb{N}}^{\mathbb{N}}$. The class of all metrizable topological groups is a proper subclass of the class $T{G}_{}$ of all topological groups having a -base. We prove that a topological group is metrizable iff it is Fréchet-Urysohn and has a -base. We also show that any precompact set in a topological group $G\in T{G}_{}$ is metrizable, and hence G is strictly angelic. We deduce from this result...

In this paper we deal with the vector lattice $C\left(B\right)$ of all elementary Carathéodory functions corresponding to a generalized Boolean algebra $B$.

Let $ℳ$ be the Banach space of real measures on a $\sigma$-ring $\mathbf{R}$, let ${ℳ}^{\text{'}}$ be its dual, let $E$ be a quasi-complete locally convex space, let $E^{\text{'}}$ be its dual, and let $\mu$ be an $E$-valued measure on $\mathbf{R}$. If is shown that for any $\theta \in {ℳ}^{\text{'}}$ there exists an element $\int \theta d\mu$ of $E$ such that $⟨x^{\text{'}}\circ \mu ,\theta ⟩=⟨\int \theta d\mu ,x^{\text{'}}⟩$ for any $x^{\text{'}}\in E^{\text{'}}$ and that the map$$\theta \to \int \theta d\mu :{ℳ}^{\text{'}}\to E$$is order continuous. It follows that the closed convex hull of $\mu \left(\mathbf{R}\right)$ is weakly compact.

Let X be a real or complex vector space. We show that the maximal p-convex topology makes X a complete Hausdorff topological vector space. If X has an uncountable dimension, then different p give different topologies. However, if the dimension of X is at most countable, then all these topologies coincide. This leads to an example of a complete locally pseudoconvex space X that is not locally convex, but all of whose separable subspaces are locally convex. We apply these results to topological algebras,...

Let $E$ be a real linear space. A vectorial inner product is a mapping from $E\times E$ into a real ordered vector space $Y$ with the properties of a usual inner product. Here we consider $Y$ to be a $ℬ$-regular Yosida space, that is a Dedekind complete Yosida space such that ${\bigcap }_{J\in ℬ}J=\left\{0\right\}$, where $ℬ$ is the set of all hypermaximal bands in $Y$. In Theorem 2.1.1 we assert that any $ℬ$-regular Yosida space is Riesz isomorphic to the space $B\left(A\right)$ of all bounded real-valued mappings on a certain set $A$. Next we prove Bessel Inequality and Parseval...

Every weakly sequentially compact convex set in a locally convex space has the weak drop property and every weakly compact convex set has the quasi-weak drop property. An example shows that the quasi-weak drop property is strictly weaker than the weak drop property for closed bounded convex sets in locally convex spaces (even when the spaces are quasi-complete). For closed bounded convex subsets of quasi-complete locally convex spaces, the quasi-weak drop property is equivalent to weak compactness....

Si studiano alcune proprietà di un certo limite induttivo di spazi non-archimedei di funzioni continue. In particolare, si esamina la completezza di questo limite induttivo e si indaga il problema di quando lo spazio coincide con il proprio inviluppo proiettivo.