IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS

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 follwing notion: a net (x ) in an LNVL (X; p;E) is said to be unbounded p-convergent to x 2 X (shortly, x upconverges to x) if p(jx 􀀀 xj ^ u) o􀀀 ! 0 in E for all u 2 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. Keywords: Vector Lattice, Lattice-Normed Vector Lattice, up-Convergence, uo-Convergence, un-Convergence, uaw-Convergence, Mixed-Normed Space