Various Aspects of Approximative τ -Compactness in Banach Spaces

We study approximative τ -compactness in Banach spaces, where τ is the norm or weak topology. The family of Banach spaces with Fr´echet differentiable norms falls under the category of spaces where every w ∗ -closed finite codimensional subspace is approximatively compact in the duals, is observed. On the other hand, we derive that if every w ∗ -closed hyperplane in X ∗ is strongly proximinal, then X is Asplund. We conclude that a smooth Banach space is Fr´echet smooth if and only if every w ∗ -closed hyperplane is approximatively compact in its dual. The property approximative τ -compactness is characterized in a variety of ways for finite codimensional subspaces. It is established that a separable Banach space X is Asplund if and only if X ∗ admits a dual CLUR renorming. This property is discussed in the context of quotient spaces. Stability results for approximative τ -compactness in the spaces of Bochner integrable functions and polyhedral sum of Banach spaces are also presented.