Let U, V be two symmetric convex bodies in ${ℝ}^n$ and |U|, |V| their n-dimensional volumes. It is proved that there exist vectors $u_1,...,u_n\in U$ such that, for each choice of signs ${\epsilon }_1,...,{\epsilon }_n=\pm 1$, one has ${\epsilon }_1{u}_1+...+{\epsilon }_n{u}_n\notin rV$ where $r={\left(2\pi e^2\right)}^{-1/2}{n}^{1/2}\left(\right|U|/{\left|V\right|)}^{1/n}$. Hence it is deduced that if a metrizable locally convex space is not nuclear, then it contains a null sequence $\left(u_n\right)$ such that the series ${\sum }_{n=1}^{\infty }{\epsilon }_n{u}_{\pi \left(n\right)}$ is divergent for any choice of signs ${\epsilon }_n=\pm 1$ and any permutation π of indices.

On étudie les bases de Schauder pour fonctions holomorphes et leurs applications à l’approximation et interpolation.Après avoir établi quelques faits généraux sur les bases et semi-bases, on les applique à l’étude des bases formées par une suite simple de polynômes.L’effort principal est porté sur la preuve de l’existence d’une “bonne” base commune des espaces des fonctions holomorphes sur $\Omega$ et $\chi$, où $\Omega$ est un domaine de $\mathbf{C}$ et $\chi$ un compact dans $\Omega$ tels que $\Omega \setminus \chi$ soit un domaine régulier pour le problème...

We prove precise decomposition results and logarithmically convex estimates in certain weighted spaces of holomorphic germs near ℝ. These imply that the spaces have a basis and are tamely isomorphic to the dual of a power series space of finite type which can be calculated in many situations. Our results apply to the Gelfand-Shilov spaces $S{¹}_{\alpha }$ and $S{₁}^{\alpha }$ for α > 0 and to the spaces of Fourier hyperfunctions and of modified Fourier hyperfunctions.

Let F be a complemented subspace of a nuclear Fréchet space E. If E and F both have (absolute) bases $\left(e_n\right)$ resp. $\left(f_n\right)$, then Bessaga conjectured (see [2] and for a more general form, also [8]) that there exists an isomorphism of F into E mapping $f_n$ to $t_n{e}_{\pi \left(k_n\right)}$ where $\left(t_n\right)$ is a scalar sequence, π is a permutation of ℕ and $\left(k_n\right)$ is a subsequence of ℕ. We prove that the conjecture holds if E is unstable, i.e. for some base of decreasing zero-neighborhoods $\left(U_n\right)$ consisting of absolutely convex sets one has ∃s ∀p ∃q ∀r $li{m}_n\left(d_{n+1}(U_q,U_p)\right)/\left(d_n(U_r,U_s)\right)=0$ where...

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...