En admettant la conjecture de Dickson, nous démontrons que, pour chaque couple d’entiers $q\>0$ et $k\>0$, il existe une partie infinie $L_{q,k}\subset \mathbb{N}$ telle que, pour chacun des entiers $n\in L_{q,k}$ et tout entier $s$ tel que $0\<\left|s\right|\le q$, on ait $n+s=\left|s\right|t_1...t_k$ où $t_1\<...\<t_k$ sont des nombres premiers. De même, pour chaque couple d’entiers $q\>0$ et $k\>0$, il existe une partie infinie $M_{q,k}\subset \mathbb{N}$ telle que, pour chacun des entiers $n\in M_{q,k}$ et tout entier $s$ (nul ou non ) de l’intervalle $\left[-q,q\right]$, on ait $n+s=l{t}_1...t_k$ où $t_1\<...\<t_k$ sont des nombres premiers et l’entier $l$ appartient à l’intervalle $\left[1,2q+1\right]$. La lecture non standard...

The purpose of this paper is to present Fatou type results for a sequence of Pettis integrable functions and multifunctions. We prove the non vacuity of the weak upper limit of a sequence of Pettis integrable functions taking their values in a locally convex space and we deduce a Fatou's lemma for a sequence of convex weak compact valued Pettis integrable multifunctions. We prove as well a Lebesgue theorem for a sequence of Pettis integrable multifunctions with values in the space of convex compact...

Let $X$ and $Y$ be vector spaces over the same field $K$. Following the terminology of Richard Arens [Pacific J. Math. 11 (1961), 9–23], a relation $F$ of $X$ into $Y$ is called linear if $\lambda F\left(x\right)\subset F\left(\lambda x\right)$ and $F\left(x\right)+F\left(y\right)\subset F(x+y)$ for all $\lambda \in K\setminus \left\{0\right\}$ and $x,y\in X$. After improving and supplementing some former results on linear relations, we show that a relation $\Phi$ of a linearly independent subset $E$ of $X$ into $Y$ can be extended to a linear relation $F$ of $X$ into $Y$ if and only if there exists a linear subspace $Z$ of $Y$ such that $\Phi \left(e\right)\in Y|Z$ for all $e\in E$. Moreover, if $E$ generates...

For a Tychonoff space $X$, $C\left(X\right)$ is the lattice-ordered group ($l$-group) of real-valued continuous functions on $X$, and $C^{*}\left(X\right)$ is the sub-$l$-group of bounded functions. A property that $X$ might have is (AP) whenever $G$ is a divisible sub-$l$-group of $C^{*}\left(X\right)$, containing the constant function 1, and separating points from closed sets in $X$, then any function in $C\left(X\right)$ can be approximated uniformly over $X$ by functions which are locally in $G$. The vector lattice version of the Stone-Weierstrass Theorem is more-or-less equivalent...