Let $X$ be an Archimedean Riesz space and $𝒫\left(X\right)$ its Boolean algebra of all band projections, and put ${𝒫}_e=\{Pe:P\in 𝒫\left(X\right)\}$ and ${ℬ}_e=\{x\in X:x\wedge (e-x)=0\}$, $e\in X^{+}$. $X$ is said to have Weak Freudenthal Property ($WFP$) provided that for every $e\in X^{+}$ the lattice $lin{𝒫}_e$ is order dense in the principal band $e^{dd}$. This notion is compared with strong and weak forms of Freudenthal spectral theorem in Archimedean Riesz spaces, studied by Veksler and Lavrič, respectively. $WFP$ is equivalent to $X^{+}$-denseness of ${𝒫}_e$ in ${ℬ}_e$ for every $e\in X^{+}$, and every Riesz space with sufficiently many projections...

The main part of the paper is devoted to the problem of the existence of absolutely representing systems of exponentials with imaginary exponents in the spaces $C^{\infty }\left(G\right)$ and $C^{\infty }\left(K\right)$ of infinitely differentiable functions where G is an arbitrary domain in ${ℝ}^p$, p≥1, while K is a compact set in ${ℝ}^p$ with non-void interior K̇ such that $\overline{K}̇=K$. Moreover, absolutely representing systems of exponents in the space H(G) of functions analytic in an arbitrary domain $G\subseteq {ℂ}^p$ are also investigated.

Extensions of order bounded linear operators on an Archimedean vector lattice to its relatively uniform completion are considered and are applied to show that the multiplication in an Archimedean lattice ordered algebra can be extended, in a unique way, to its relatively uniform completion. This is applied to show, among other things, that any order bounded algebra homomorphism on a complex Archimedean almost $f$-algebra is a lattice homomorphism.

Let X, Y be two separable F-spaces. Let (Ω,Σ,μ) be a measure space with μ complete, non-atomic and σ-finite. Let $N_F$ be the Nemytskiĭ set-valued operator induced by a sup-measurable set-valued function $F:\Omega \times X\to 2^Y$. It is shown that if $N_F$ maps a modular space $(N\left(L(\Omega ,\Sigma ,\mu ;X)\right),{\varrho }_{N,\mu })$ into subsets of a modular space $(M\left(L(\Omega ,\Sigma ,\mu ;Y)\right),{\varrho }_{M,\mu })$, then $N_F$ is automatically modular bounded, i.e. for each set K ⊂ N(L(Ω,Σ,μ;X)) such that $r_K=sup{\varrho }_{N,\mu }\left(x\right):x\in K<\infty$ we have $sup{\varrho }_{M,\mu }\left(y\right):y\in N_F\left(K\right)<\infty$.

In this paper it is proved that if $\{E_n{\}}_{n=1}^{\infty }$ and $\{F_n{\}}_{n=1}^{\infty }$ are two sequences of infinite-dimensional Banach spaces then $H=\left({\oplus }_{n=1}^{\infty }{E}_n\right)\times {\prod }_{n=1}^{\infty }{F}_n$ is not $B_r$-complete. If $\{E_n{\}}_{n=1}^{\infty }$ and $\{F_n{\}}_{n=1}^{\infty }$ are also reflexive spaces there is on $H$ a separated locally convex topology $ℱ$, coarser than the initial one, such that $H\left[ℱ\right]$ is a bornological barrelled space which is not an inductive limit of Baire spaces. It is given also another results on $B_r$-completeness and bornological spaces.