Unbounded p-convergence in lattice-normed vector lattices

Marabeh, Mohammad A. A.
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 $(x_alpha)$ in an LNVL $(X,p,E)$ is said to be unbounded $p$-convergent to $x in X$ (shortly, $x_alpha$ $up$- converges to $x$) if $p(lvert x_alpha −x rvert wedge u) xrightarrow{o}0$ 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.