Various aspects of approximative \(\tau \)-compactness in Banach spaces

We study approximative \(\tau \) $\tau$ -compactness in Banach spaces, where \(\tau \) $\tau$ is the norm or weak topology. The family of Banach spaces with Fréchet differentiable norms falls under the category of spaces where every \(w^*\) $w^{\ast }$ -closed finite codimensional subspace is approximately compact in the duals, is observed. On the other hand we derive that if every \(w^*\) $w^{\ast }$ -closed hyperplane in \(X^*\) $X^{\ast }$ is strongly proximinal then X is Asplund. We conclude that a smooth Banach space is Fréchet smooth if and only if every \(w^*\) $w^{\ast }$ -closed hyperplane is approximatively compact in its dual. The property approximative \(\tau \) $\tau$ -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 \(X^*\) $X^{\ast }$ admits a dual CLUR renorming. This property is discussed in the context of quotient spaces. Stability results for approximative \(\tau \) $\tau$ -compactness in the spaces of Bochner integrable functions and polyhedral sum of Banach spaces are also presented.