Let X be Banach space. For \(x\in X\) with \(\Vert x\Vert =1\) , we denote the state space by, \(S_{x}=\{x^*\in X^*:\Vert x^*\Vert =x^*(x)=1\}\) . In this paper, we study \(\hbox {weak}^*\) -weak and \(\hbox {weak}^*\) - \(\Vert .\Vert \) points of continuity of the identity map, on the state spaces in the space \(\ell ^{p}(X)\) for \(1<p<\infty \) for a non-reflexive Banach space X and then we use these results to characterize the weak and norm compactness of the state spaces of unit vectors in \(\ell ^{p}(X)\) . In addition, we address an open problem, the characterization of weakly compact state spaces in the space of Bochner-integrable functions \(L^{1}(\mu , X)\) (see Daptari and Dwivedi in Colloq Math 179(1):87–105, 2025, Problem 3.18). We also provide a local solution to this problem without any additional assumptions on the Banach space X. Motivated by the work of Daptari et al. (Rev R Acad Cienc Exactas Fís Nat Ser A Mat 119(3):82, 2025), we show that if the set of all \(\hbox {weak}^{*}\) -weak points of continuity of \(L^{1}(\mu , X)_{1}^{*}\) is weakly dense in \(L^{1}(\mu , X)_{1}^{*}\) , then \(X^{*}\) has the Radon–Nikodým property (RNP) (see Theorem 3.12).
