In this paper, we shall give the first and second $\alpha$-, $\beta$-, $\gamma$-, and $f$- duals of the sets $w^p_0$, $w^p$, and $w^p_\infty$ of sequences that are strongly summable and bounded with index $p$ by the Cesáro method $C_1$, and of the sets $c_0(\Lambda)$, $c(\Lambda)$ and $c_\infty(\Lambda)$ of sequences that are $\Lambda$-strongly convergent to naught, convergent and bounded. Furthermore, we shall determine the first and second continuous dual spaces of the sets $w^p_0$, $w^p$, $c_0(\Lambda)$ and $c(\Lambda)$. Finally we give necessary and sufficient conditions for an infinite matrix to be a map between any two of these spaces and apply the Hausdorff measure of noncompactness to give necessary and sufficient conditions for a matrix to be a compact linear operator between these spaces.