Function space with $\|\cdot\|_g$ norm

Question

I'm reading some papers that utilize functional analysis. I came across the following definition: Let $\mathcal{X}$ be a metric space and let $g:\mathcal{X}\rightarrow\mathbb{R}$ be a continuous, strictly positive function. We define $C_g(\mathcal{X})$ to be the space of all continuous $f:\mathcal{X}\rightarrow \mathbb{R}$ such that $\|f\|_g<\infty$, where $$\|f\|_g := \sup_{x\in\mathcal{X}} \,\frac{\vert f(x)\vert}{g(x)}.$$

I am looking for a reference that talks about this space in more detail. I couldn't find anything with a quick google search. (I'm not even sure if this space has a name.) Apparently $C_g(\mathcal{X})$ is a Banach space w.r.t. the norm $\|\cdot\|_g$. But I would like to understand this space better.

Answer 1

Your space $C_g(\mathcal{X})$ is isometric to a more standard Banach space $C_b(\mathcal{X})$, the space of bounded continuous functions on $\mathcal{X}$, where the correspondence is \begin{align*} f\in C_g(\mathcal{X})&\longrightarrow \frac{f}{g}\in C_b(\mathcal{X})\\ gf\in C_g(\mathcal{X})&\longleftarrow f\in C_b(\mathcal{X}). \end{align*}