On spaces in which the norm convergence coincides with the order convergence

Order convergence is not topological in infinite-dimensional vector lattices

“…As an extension of the work in [5,6,15], Chuchaev investigated ordered locally convex spaces, where the topological convergence agrees with the order convergence of eventually topologically bounded nets; see, for example, Theorem 2.3, Propositions 2.4, 2.5, and 2.6 in [4]. In addition, he studied ordered locally convex spaces, where the topological convergence is equivalent to the order convergence of sequences; see, for example, Propositions 3.2, 3.3, 3.4 and 3.5 in [4].…”

“…In [15,Theorem 1], the authors characterized ordered normed spaces in which the order convergence of nets coincides with the norm convergence. Also, they characterized ordered normed spaces in which order convergence and norm convergence coincide for sequences; see [15,Theorem 3].…”

“…It should be noticed that Theorem 1 was proven in the case of ordered normed spaces with minihedral cones (see [15,Theorem 2]), in the case of Banach lattices (see [16]), and in the case of normed lattices (see [9,Theorem 2]). A useful characterization of uo-convergence in atomic vector lattices is given in Proposition 1.…”

“…In this paper, our main result is that if (X, τ ) is a topological vector lattice such that x α τ − → 0 iff x α o − → 0 for any net x α in X, then dim X < ∞; see Theorem 1. It should be noticed that Theorem 1 was proven in the case of ordered normed spaces with minihedral cones (see [15,Theorem 2]), in the case of Banach lattices (see [16]), and in the case of normed lattices (see [9,Theorem 2]). A useful characterization of uo-convergence in atomic vector lattices is given in Proposition 1.…”