Weight generalization of the space of continuous functions vanishing at infinity

Reza Saleki and Hojjatollah Samea

Mathematische Nachrichten, 2023, vol. 296, issue 8, 3619-3629

Abstract: 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‐negative function) on X. We give a correct and comprehensive definition of the weighted generalization C0ν(X)$C_0^\nu (X)$ of C0(X)$C_0(X)$, and show that it is a seminormed space with respect to the canonical seminorm ∥f∥ν=supx∈X|f(x)|$\Vert f\Vert _\nu =\sup _{x\in X}|f(x)|$, where f∈C0ν(X)$f\in C_0^\nu (X)$. We find conditions on ν under which C0ν(X)$C_0^\nu (X)$, with respect to ∥.∥ν$\Vert .\Vert _\nu$, becomes a normed space or a Banach space or an algebra, or a topological algebra, respectively.

Date: 2023
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https://doi.org/10.1002/mana.202200021

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