Nonstandard hulls of lattice-normed ordered vector spaces

Abstract

Nonstandard hulls of a vector lattice were introduced and studied in many papers. Recently, these notions were extended to ordered vector spaces. In the present paper, following the construction of associated Banach-Kantorovich space due to Emelyanov, we describe and investigate the nonstandard hull of a lattice-normed space, which is the foregoing generalization of Luxemburg's nonstandard hull of a normed space.

Compact-like operators in lattice-normed spaces

“…If X is a vector lattice and the vector norm p is monotone (i.e. |x| ≤ |y| ⇒ p(x) ≤ p(y)) then the triple (X, p, E) is called a lattice-normed vector lattice, abbreviated as LN V L; see [5,6,7].…”

Unbounded p-Convergence in Lattice-Normed Vector Lattices

“…(4) If X is op τ -continuous, then clearly every weak unit of X is a p τ -unit. (5) In an LSN V L (E, |·|, E τ ) with (E, τ ) having the Lebesgue property, the lattice norm p(x) = |x| is always op τ -continuous. Therefore, the notions of p τ -unit and of weak unit coincide in E.…”

“…

Let (x α ) be a net in a vector lattice normed by locally solid lattice (X, p, E τ ). We say that (This convergence has been studied recently for lattice-normed vector lattices as the up-convergence in [5,6,7], the uo-convergence in [14], and, as the un-convergence in [10,13,14,16,18]. In this paper, we study the general properties of the unboundedLet X be a vector space, E be a vector lattice, and p : X → E + be a vector norm (i.e.

…”

The statistically unbounded τ-convergence on locally solid Riesz spaces

“…e.g. [4][5][6][7]). However, as far as we know, the concept of unbounded order convergence related to the statistical convergence has not been done before.…”