Let $L$ be a completely regular frame and $mathcal{R}L$ be the ‎ring of continuous real-valued functions on $L$‎. ‎We show that the‎ ‎lattice $Zid(mathcal{R}L)$ of $z$-ideals of $mathcal{R}L$ is a‎ ‎normal coherent Yosida frame‎, ‎which extends the corresponding $C(X)$‎ ‎result of Mart'{i}nez and Zenk‎. ‎This we do by exhibiting‎ ‎$Zid(mathcal{R}L)$ as a quotient of $Rad(mathcal{R}L)$‎, ‎the‎ ‎...

In this paper, we first introduce some function spaces, with certain locally convex topologies, closely  related to the space of real-valued continuous functions on $X$,  where $X$ is a $C$-distinguished topological space. Then, we show that their dual spaces can be identified in a natural way with certain spaces of Radon measures.

In pointfree topology, a continuous real function on a frame L is a map L(R) → L from the frame of reals into L. The discussion of continuous real functions with possibly infinite values can be easily brought to pointfree topology by replacing the frame L(R) with the frame of extended reals L ( R ) (i.e. the pointfree counterpart of the extended real line R = R ∪ {±∞}). One can even deal with a...

In this note we present a new treatment of the pointfree version of the semicontinuous quasi-uniformity based on the new tool of the ring of arbitrary (not necessarily continuous) real-valued functions made available recently by J. Gutiérrez Garćıa, T. Kubiak and J. Picado [Localic real functions: a general setting, Journal of Pure and Applied Algebra 213 (2009) 1064-1074]. The purpose is to sh...

In this paper, we will be working with rings and ideals of real analytic functions. Given x ∈ R, let Ax be the ring of real-valued functions f : R → R that are analytic at x. When x = 0 is the origin, it is useful to think of A0 as a subring of the formal power series ring R[[ω1, . . . , ωd]] = R[[ω]] which consists of power series which are convergent in some neighborhood of the origin. For al...

In the early sixties, Amice ([Am1], [Am2]) studied the space of K-valued, locally analytic functions on Z Z p and formulated a complete description of its dual, the ring of K-valued, locally Q p-analytic distributions on Z Z p , when K is a complete subfield of C p. She found an isomorphism between the ring of distributions and the space of global functions on a rigid variety over K parameteriz...

Although prox-regular functions in general are nonconvex, they possess properties that one would expect to find in convex or lowerC2  functions. The class of prox-regular functions covers all convex functions, lower C2  functions and strongly amenable functions. At first, these functions have been identified in finite dimension using proximal subdifferential. Then, the definition of prox-regula...