Local condition as a crucial tool for local structure of Calderón–Lozanovskiĭ spaces

We shall give a full criterion for a point x ∈ E φ to be a point of order continuity of Calderón–Lozanovskiĭ spaces E φ . In order to get that result we introduce the new local Δ 2 E ( x ) condition which appears to be necessary and sufficient for this goal. Next we study the structure of points of lower local uniform monotonicity (LLUM for short). We give a result concerning general Köthe sequence spaces and we present a characterization for LLUM points in Calderón–Lozanovskiĭ spaces.