The talk presented a survey of results most of which have been obtained over the last several years in collaboration with M.Florencio and P.J.Paúl (Seville). The results concern the question of barrelledness of locally convex spaces equipped with suitable Boolean algebras or rings of projections. They are particularly applicable to various spaces of measurable vector valued functions. Some of the results are provided with proofs that are much simpler than the original ones.

Let ${\left(E_i\right)}_{i\in I}$ be a family of normed spaces and $\lambda$ a space of scalar generalized sequences. The $\lambda$-sum of the family ${\left(E_i\right)}_{i\in I}$ of spaces is $$\lambda \left\{{\left(E_i\right)}_{i\in I}\right\}:=\{{\left(x_i\right)}_{i\in I},x_i\in E_i,\text{and}(\parallel x_i{\parallel )}_{i\in I}\in \lambda \}.$$ Starting from the topology on $\lambda$ and the norm topology on each $E_i,$ a natural topology on $\lambda \left\{{\left(E_i\right)}_{i\in I}\right\}$ can be defined. We give conditions for $\lambda \left\{{\left(E_i\right)}_{i\in I}\right\}$ to be quasi-barrelled, barrelled or locally complete.

As a counterpart to classical topological vector spaces in the alternative set theory, biequivalence vector spaces (over the field $Q$ of all rational numbers) are introduced and their basic properties are listed. A methodological consequence opening a new view towards the relationship between the algebraic and topological dual is quoted. The existence of various types of valuations on a biequivalence vector space inducing its biequivalence is proved. Normability is characterized in terms of total...

Soient $a_1,...,a_n$ des éléments d’une $b$-algèbre commutative unifère $A$. On définit et étudie un “spectre” de $a=(a_1,...,a_n)$ qui dépend de la croissance des fonctions $u_1\left(s\right),...,u_n\left(s\right)$ de l’égalité spectrale$$(a_1-s_1)u_1\left(s\right)+\cdots +(a_n-s_n)u_n\left(s\right)=1$$près du spectre simultané. À partir des propriétés de ce spectre, on construit un calcul fonctionnel qui, réduit au cas banachique, s’étend à certaines fonctions supposées seulement holomorphes à l’intérieur du spectre simultané. Ce calcul fonctionnel permet aussi d’étudier la régularité des éléments $a_1,...,a_n$ et des fonctions $u_1\left(s\right),...,u_n\left(s\right)$.