In this paper, we establish some density theorems in the setting of particular locally convex vector lattices of continuous functions de ned on a locally compact Hausdorff space, which we introduced and studied in [3,4] and which we named regular vector lattices. In this framework, by using properties of the subspace of the so-called generalized af ne functions, we give a simple description of the closed vector sublattice, the closed Stone vector sublattice and the closed subalgebra generated by...

Two problems posed by Choquet and Foias are solved:(i) Let $T$ be a positive linear operator on the space $C\left(X\right)$ of continuous real-valued functions on a compact Hausdorff space $X$. It is shown that if $n^{-1}{\sum }_{r=0}^{n-1}{T}^{r}1$ converges pointwise to a continuous limit, then the convergence is uniform on $X$.(ii) An example is given of a Choquet simplex $K$ and a positive linear operator $T$ on the space $A\left(K\right)$ of continuous affine real-valued functions on $K$, such that$$\inf \left\{\right(T^{n}1\left)\right(x):n\ge \}\<1$$for each $x$ in ${\partial }_{\ell }K$, but $\parallel T^{n}1\parallel$ does not converge to 0.

The aim of this note is 1. to show that some results (concerning the structure of the solution set of equations (18) and (21)) obtained by Czarnowski and Pruszko in [6] can be proved in a rather different way making use of a simle generalization of a theorem proved by Vidossich in [8]; and 2. to use a slight modification of the “main theorem” of Aronszajn from [1] applying methods analogous to the above mentioned idea of Vidossich to prove the fact that the solution set of the equation (24), (25)...

Let $K$ be a compact space and let $C\left(K\right)$ be the Banach lattice of real-valued continuous functions on $K$. We establish eleven conditions equivalent to the strong compactness of the order interval $[0,x]$ in $C\left(K\right)$, including the following ones: (i) $\{x>0\}$ consists of isolated points of $K$; (ii) $[0,x]$ is pointwise compact; (iii) $[0,x]$ is weakly compact; (iv) the strong topology and that of pointwise convergence coincide on $[0,x]$; (v) the strong and weak topologies coincide on $[0,x]$. Moreover, the weak topology and that of pointwise convergence...