In this paper, we characterize the weighted generalization of the space of continuous functions vanishing at infinity and correct some wrong results in the paper. Let 𝑋 be a locally compact space and 𝜈 is an arbitrary weight (non-negativefunction) on 𝑋. We give a correct and comprehensive definition of the weightedgeneralization 𝐶𝜈0 (𝑋) of 𝐶0(𝑋), and show that it is a seminormed space withrespect to the canonical seminorm ‖𝑓‖𝜈 = sup𝑥∈𝑋 |𝑓(𝑥)|, where 𝑓∈𝐶𝜈0 (𝑋). Wefind conditions on 𝜈 under which 𝐶𝜈0 (𝑋), with respect to ‖.‖𝜈, becomes a normedspace or a Banach space or an algebra, or a topological algebra, respectively.
