Unbounded p-Convergence in Lattice-Normed Vector Lattices

Abstract

The main aim of this thesis is to generalize unbounded order convergence, unbounded norm convergence and unbounded absolute weak convergence to lattice-normed vector lattices (LNVLs). Therefore, we introduce the following notion: a net (x_α) in an LNVL (X; p;E) is said to be unbounded p-convergent to x ∈ X (shortly, x_α up-converges to x) if p(|x_α- x|^ u) o→ 0 in E for all u ∈ X+. Throughout this thesis, we study general properties of up-convergence. Besides, we introduce several notions in lattice-normed vector lattices which correspond to notions from vector and Banach lattice theory. Finally, we study briefly the mixed-normed spaces and use them for an investigation of up-null nets and up-null sequences in lattice-normed vector lattices.