The familiar classical result that a continuous map from a space $X$ to a space $Y$ can be defined by giving continuous maps $varphi_U: U to Y$ on each member $U$ of an open cover ${mathfrak C}$ of $X$ such that $varphi_Umid U cap V = varphi_V mid U cap V$ for all $U,V in {mathfrak C}$ was recently shown to have an exact analogue in pointfree topology, and the same was done for the familiar cla...

where 0 < T <∞; φ ∈ 0 := C([−τ,0];X), τ > 0; the linear operator A, defined from the domain D(A) ⊂ X into X , is such that −A generates a C0 semigroup {S(t) : t ≥ 0} of bounded linear operators in X ; the nonlinear map f is defined from [0,T]× T , T := C([−τ,T];X), into X ; the map h is defined from 0 into 0; and for (t,u) ∈ [0,T]× T , the function ut ∈ 0 is given by ut(s) = u(t + s) for s ∈ [−...

This paper is the first in a series of papers developing a functionalanalytic theory of vertex (operator) algebras and their representations. For an arbitrary Z-graded finitely-generated vertex algebra (V, Y,1) satisfying the standard grading-restriction axioms, a locally convex topological completion H of V is constructed. By the geometric interpretation of vertex (operator) algebras, there is...

In this paper, we first characterize the continuity of a map in space $ \mathcal{C} = BC(\mathcal{H},\mathbb{R}^m) equipped with compact open topology. Then show that linear lattice and nonlocal dispersal equations generate uniformly continuous semigroups Banach \mathcal{B} supremum norm. Finally, illustrate how to prove nonlinear monotone semiflows respect

Let E and F be Banach spaces. It is proved that if Ω and Ω are open subsets of R and R , respectively, and T is a linear biseparating map between two spaces of differentiable functions A(Ω, E) and A(Ω, F ), then p = q, n = m, and there exist a diffeomorphism h of class C from Ω onto Ω, and a map J : Ω → L(E, F ) of class s−C such that for every y ∈ Ω and every f ∈ A(Ω, E), (Tf)(y) = (Jy)(f(h(y)...

A completely n -positive linear map from a locally C∗-algebra A to another locally C∗-algebra B is an n × n matrix whose elements are continuous linear maps from A to B and which verifies the condition of completely positivity. In this paper we prove a Radon-Nikodym type theorem for strict completely n-positive linear maps which describes the order relation on the set of all strict completely n...