Uniform K-monotonicity and K-order continuity in symmetric spaces with application to approximation theory.

We investigate K -order continuity in a symmetric space E using the fundamental function ϕ of E . We also show a connection between reflexivity and K -order continuity in symmetric spaces. Next, we present several results devoted to a characterization of uniform K -monotonicity and decreasing (increasing) uniform K -monotonicity in symmetric spaces. We also discuss a relationship between decreasing (resp. increasing) uniform monotonicity and decreasing (resp. increasing) uniform K -monotonicity. Next, we deliberate a correlation between uniform K -monotonicity and uniform rotundity in symmetric spaces. Finally, employing K -monotonicity properties and K -order continuity we provide solvability and stability of the best approximation problem in the sense of the Hardy–Littlewood–Pólya relation ≺ in symmetric spaces. [ABSTRACT FROM AUTHOR]