Extremity in Köthe–Bochner Function Spaces

LetEbe a Köthe function space over a complete measurable space andXa Banach space. Recall an elementhinEis said to beorder continuousif, for any decreasing sequence {gn} inSE, ⋀ngn=0 andgn≤|h| implies limn→∞gn=0. We show that every denting point of the unit ball ofEis order continuous. Using this result, we prove thatfis a denting point of the unit ball ofE(X) if and only if•‖(·)‖is a denting point of the unit ball of.•for almost all∈supp,()/‖()‖is a denting point of the unit ball of.Suppose thatEis order continuous. We also prove that for any unit vectorfinE(X), if ‖f(·)‖Xis a strongly exposed point of the unit ball ofEand for almost allt∈suppf,f(t)/‖f(t)‖Xis a strongly exposed point of the unit ball ofX, thenfis a strongly exposed point of the unit ball ofE(X).