A Characterization of Norm Compactness in the Bochner Space $L^p(G;B)$ For an Arbitrary Locally Compact Group $G$

In this paper, we generalize a result of N. Dinculeanu which characterizes norm compactness in the Bochner space $L^p(G;B)$ in terms of an approximate identity and translation operators, where $G$ is a locally compact abelian group and $B$ is a Banach space. Our characterization includes the case where $G$ is nonabelian, and we weaken the hypotheses on the approximate identity used, providing new results even for the case $B=\mathbb{C}$ and $G = $\mathbb{R}$^n.$