Compact-Like Operators in Vector Lattices Normed by Locally Solid Lattices

Lattice-valued norms on vector lattices play an important and effective role in the functional analysis. This chapter introduces continuous and bounded operators with respect to the p_τ-convergence, and the compact operators in vector lattice normed by locally solid lattices. The concept of the unbounded convergence is crucial for the concept of p_τ-convergence. The chapter presents some basic notions and properties of vector lattices, lattice-normed spaces, locally solid Riesz spaces, unbounded convergence, and unbounded p_τ-convergence. It outlines the notions of up_τ-continuous and sequentially up_τ-continuous operator between lattice-normed locally solid vector lattices. The chapter shows that a sequence of order-bounded sequentially p_τ-compact operators is sequentially p_τ-compact. Also, it holds for the equicontinuously and uniformly convergence.