Let E be a dual Banach space.
E is said to have quasi-weak∗-normal structure if for each weak∗ compact convex
subset K of E there exists x ∈ K such that ∥x − y∥ <diam(K) for all y ∈ K. E is
said to satisfy Lim’s condition if whenever {xα} is a bounded net in E converging to
0 in the weak∗ topology and lim∥xα∥ = s then limα∥xα+ y∥ = s + ∥y∥ for any
y ∈ E. Lim’s condition implies (quasi) weak∗-normal structure. Let H be a Hilbert
space. In this paper, we prove that 𝒯 (H), the space of trace class operators on H,
always has quasi-weak∗-normal structure for any H; 𝒯 (H) satisfies Lim’s
condition if and only if H is finite dimensional. We also prove that the space
of bounded linear operator on H has quasi-weak∗-normal structure if and
only if H is finite dimensional; the space of compact operators on H has
quasi-weak-normal structure if and only if H is separable. Finally we prove that if X
is a locally compact Hausdorff space, then C0(X)∗ satisfies Lim’s condition if and
only if C0(X)∗ is isometrically isomorphic to l1(Γ) for some non-empty set
Γ.
