  In this current work, we revisit the recent improvement of the discrete
Hardy's inequality in one dimension and establish an extended improved discrete
Hardy's inequality with its optimality. We also study one-dimensional discrete
Copson's inequality (E.T. Copson, \emph{Notes on a series of positive terms},
J. London Math. Soc., 2 (1927), 9-12.), and achieve an improvement of the same
in a particular case. Further, we study some fundamental structures such as
completeness, K\"{o}the-Toeplitz duality, separability, etc. of the sequence
spaces which originated from the improved discrete Hardy and Copson
inequalities in one dimension.
