We study approximative $\tau$-compactness in Banach spaces, where $\tau$ is the norm or weak topology. The family of Banach spaces with Fr{\'e}chet differentiable norms falls under the category of spaces where every $w^*$-closed finite codimensional subspace is approximately compact in the duals, is observed. We conclude that a smooth Banach space is Fr{\'e}chet smooth if and only if every $w^*$-closed hyperplane is approximatively compact in its dual.
The property approximative $\tau$-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 $X^*$ admits a dual $CLUR$ renorming. This property is discussed in the context of quotient spaces. We prove that, under certain conditions, the property of approximative $\tau$-compactness passes through $M$-ideals. Stability results for approximative $\tau$-compactness in the spaces of Bochner integrable functions are also presented.